Would improvised motor vehicles be feasible a century after the fall of industrial civilization?

No, at least not sustainably.

After a century, you might have both machine tools and decent steel left. Ignoring those, constructing something like a diesel or gasoline engine requires the forging of cylinders and cylinder heads that will stand high temperatures, of drive shafts to take the stresses, etc. This is, almost by definition, a technology that would also allow steam engines, spinning jennies, drop hammers, etc.

Is industrialization technology or organization?

If one thinks or talks about industrialization, the two are usually jumbled together. The image is about proud, independent guild masters, journeymen, and apprentices being replaced by factory workers under the thumb of exploiting capitalists. This is only partially correct, there were eras and sectors where capitalist organization was coupled with hand tools.

In a worldbuilding sense, one might envision early industrial technology without the social organization -- village smiths teaching their apprentices how to turn a cylinder head on a lathe. But at some point, one needs scale to make things work. Iron may be smelted by a blacksmith, but a blast furnace needs large numbers of workers.

A century after an apocalypse event, the biggest problem with any attempt to build machinery is the memory of what was. People will try to recreate what was, such as buggies and motorcycles, when the background supply chain doesn't exist. Most importantly, politicians will lead people in "cargo cult" type activities building "almost machines" which will waste resources.

This question greatly underestimates the social and emotional impact of an apocalypse event. It can take hundreds of years to rebuild society so that someone could explore building machines.

When the population is in small 500 person or less villages (and far more will be in the "less" scattered all over the countryside), people will be focused on survival. Mechanical toys can be built when there is excess food production and relative stability, not when people are scrabbling for food and fighting off raids. For example, look at the Yanomami tribe. During one study, a village experienced a raid about once every 3 weeks. Another study found that the death rate from violence was more than what our soldiers experienced in WWII. During such a situation, people do not have the excess time, emotional stability, and social support for exploring building mechanical equipment that will actually work.

If you are very lucky, someone will realize that simple machines built out of local materials are possible. These machines will solve a local problem - most likely one for either growing more crops or harvesting them.

I can't remember where I heard this, but I remember someone saying that you can have steam powered trains running almost indefinitely - because all you need is blacksmithing tools (assuming no catastrophic failures, I guess). While the complexity of diesel components requires parts made with specialty tooling and in-depth experience with a certain level of technology (plus supply chains, etc.). Modern gasoline engines, of course, are very reliant on computers and mechanical parts with high levels of precision - plus the gasoline.

I live in a rural area and Antique Tractor shows are routine, with tractors having to be 50+ years old to even be considered antique. They have a reputation of being very reliable and from my understanding they make minimal use of rubber, plastic and electronics. Some of the WWII era tractors used iron wheels because of rationing. The problem is going to be how to power them, as ethanol requires a large amount of grain (that could be otherwise used to feed people). Steam might be impractical for vehicles, because of fuel and water storage needs.

What could happen is older farm equipment being used - not as transportation - but for power generation. They could be converted to steam and then used to power grist mills, lumber mills, etc. Maybe most villages have one old tractor that slowly becomes less and less tractor like, as primitive repairs modify them into tools. They could become almost impossible to move, so not something that raiders would be able to steal.

Also, if the rail system isn't completely wrecked, it may even be possible to find and repair some antique steam trains to use as actual transportation. I would imagine that this would be somewhat rare and more likely to be found between better-off communities where there are usable tracks between them. But these would be vulnerable to roving bandits and the like.

Another potential non-motorized transport system could be horse-drawn trains. It looks like this was used in history, so it seems reasonably practical - if perhaps very limited in capacity and endurance.

100 years will not destroy the world's supply of vehicles.

Quite a lot of pre-apocalypse vehicles are likely to survive the apocalypse, not intact, but with key parts such as the engine block still usable. The US military stores fighter jets sitting in a dry desert - much more fragile than an old car engine.

100 year old cars are routinely being found forgotten somewhere and restored - just search for "100 year old car barn" and videos will pop up. In a hot wet climate, everything will turn to rust. In permafrost or in desert climates, metal objects can last for a very long time.

Car bodies are carbon or low-alloy steel less than 1mm thick, so they won't be driveable, and modern cars dependent on computer ECU will likely be beyond salvage. But engines are thick aluminium and high-alloy steel, resistant to rust. Old carbureted engines will need replacement parts, especially anything non-metal, but many might still be repairable.

Restored engines, running on low-quality fuels, will not perform to their original specifications. But even tuned down to 1/10 of their original power, they'll still be better than whatever engine you can build new without large-scale industry.

Diesel engines will be strongly preferable, as diesel stores better than gasoline, and they can be run on non-petroleum fuels once that runs out.

The vehicles you'd build around these engines, in a society this basic, are likely to be closer to powered carts or primitive tractors, rather than buggies or motorcycles.

For hot wet climates, agrarian societies might actually prefer animals to vehicles, since grazing grounds will be plentiful, and the terrain difficult to traverse on wheels.

I doubt very much that in the situation you describe that there would be any ability to create motorised transport from scratch.

With a population size of just 500 and limited access to much in the way of technology everyone will be drawn into farming. Without mechanisation they will be reduced to scythes, ploughing with horses and many more very labour intensive jobs. They will probably have their hands full just growing enough food unless they are in a very agriculturally friendly environment.

Those not employed in the fields would be spending a lot of their time making or repairing tools, maintaining buildings, milling floor, baking, cooking, looking after children and animals, repairing fences, cutting wood etc.

Nobody is going to have much time for smelting iron, casting cylinder blocks or trying to make pistons or crankshafts and they would probably not be able to do any metal drilling, milling or other machining without the right equipment and machine tools and more importantly without the inclination due to the overriding importance of food production.

Any remnants of iron and steel would probably be scavenged and turned into tools. Without trade such small isolated groups would slowly degenerate to very primitive life style using materials immediately to hand and with out genetic flow in the form of moving populations they would become inbred and would probably die out.

With trade and genetic flow between settlements there might be sufficient resource to continue in such a pre-industrial society indefinitely. But if every time a settlement grows beyond a certain limit it is destroyed, they would never be able to industrialise and might well develop means to minimise or control their population level for fear of the destruction it would bring.

Totally Feasible

This is not only feasible, but is also quite likely to happen in the aftermath of a The Pockyclypse. Since you specify smallish agrarian communities, I'd just like to note that we (mostly urban and suburban types) so frequently underestimate what country folks know, remember and are capable of.

I'm seeing a lot of relatively negative answers, when I believe the truth of the matter is that machine culture in your scenario will not only survive but thrive within and past the century mark of your proposal. The truth of the matter is that "improvised motor vehicles" are already a thing. You tube is most instructive in this matter! There are so many ancient tractors (steam, petrol, diesel), engines, and other pieces of useful machinery sitting in barns, workshops, rural industrial places, garages and so forth that the people who are already in agrarian communities will have no trouble making improvised vehicles.

Will these vehicles have all the modern comforts and gadgets? No. But you don't really need any of that. A frame, four wheels, a motor, a transmission, a gear box and a steering mechanism is about all you need. Add a choke and throttle and a crank to start it up!

A real life improvised motor vechicle.

Improvised steam powered buggy and picnic lunch.

Destilling Fuel (crude to kerosene, petrol, diesel).

You may be surprised to hear that one type of improvised motor vehicle which would be both feasible and have a strong economic case in your setting is ultralight aircraft.

The economic case is scouting. Any means of detecting these roving bands of invaders in advance would be very valuable, and an ultralight aircraft can scout a large area quickly.

As far as feasibility goes, ultralight aircraft is very undemanding. In many respects you can think of it as a very big kite with an engine attached. I would expect that engine to be an air-colled two-stroke, linked directly to the propeller. Because ultralight aircraft is so light compared to the size of its wing, it can take off at surprisingly low speeds, and therefore does not need a large runway. The engine is the only part which must be made of metal; the frame and the propeller can be made out of wood (which is both lighter and readily available), and the wing out of cloth (which even the small settlements will produce to make clothes). The metal for the engine can be easily recycled; a thoroughly rusted husk of a pre-apocalypse vehicle is effectively a lump of metal oxide, and liberating metal from an oxide is a problem solved literally in prehistory.

Engine parts can be made using hand-operated tools, and finished using just two machines, both of which can be powered by human labor: a lathe and a drill. This is because in the bare-bones engine that you are producing, only the cylinder, the piston, and the intake and exhaust valves must be made to tight tolerances. It would take much longer though, so be patient.

I think the biggest problem would actually be a seal. A post-apocalypse society would probably need to create the engine block in two halves, which would then have to be put together in a way that makes the combustion chamber as close to airtight as possible. That seal must then withstand being in contact with hot combustion gases, at the very least for the duration of the flight. I'm not sure what material would have these properties and also be reliably available post-apocalypse.

Small ordinary settlements are unlikely to be able to support such technical understanding and skills.

But what about eg. monasteries that are located in old factories? People who devote themselves to technology and make their living by providing engines and services could be feasible.

Instead of threatening your reader's suspension of disbelief, simply use an existing technology that is more likely to last.

The Bicycle

A bike is not a precision mechanical instrument like an engine, and can operate with some amazingly sloppy tolerances.

Lubrication doesn't need to be high quality as long as it exists. I've heard of someone lubricating a chain with mashed banana successfully, so any oily plant residue will work.

There are two main parts which will suffer with age no matter how well it has been stored. Anything with rubber in it and anything with cloth/foam (saddle/padding)

The saddle can be replaced with upholstery using new cloth and leather or a well-carved piece of wood. Tyres are the biggest challenge, but bikes have been used with solid rubber, and even with wooden or steel tyres in the past, should pneumatics be forgotten.

And in your post-apocalyptic world, farming will benefit - this monster needs nothing more than a welder and steel and creativity, and is powered eventually by the food you grow.

https://youtu.be/fVQxlnaFr0s

takes place about a century after an apocalypse event ... any pre-apocalypse vehicles would be rusted away and unusable, and gasoline would have rotted...

The one thing you didn't address, and maybe you're open to playing around with is: What was the technology like before the apocalypse? If we imagine a world with more advanced technology than today, it's easy to imagine how a fallen society might be able to make use of it later on.

Some things you could add:

• Hermetically-sealed warehouses -- Imagine a warehouse that's fallen into a crack in the earth, and acts as a dungeon to explore; when explorers reach the end there are perfectly preserved items of "ancient technology".
• Technology built to last -- If we remove the notion of "planned obsolescence" from the technology, it's quite easy to have things that last a hundred years and keep working. Maybe 99% of the tech has rusted away, but 1% was built out of high-quality materials.
• Simple interfaces -- One thing advancing technology helps with is making devices easier to use. Imagine tech with voice activation, simple UI touchscreens, etc. Forget how the device actually works? No problem!
• Renewable energy -- Devices could be powered by solar-panel paint, micro fusion engines, etc.

If you combine these concepts it's not hard to get to compact, easy-to-use industrial 3D printers, blast furnaces, auto-miners, etc. Since it's based on fictional future tech and a certain level of brokenness, you can easily tune them to achieve exactly what you want them to.

Not the way you describe "rudimentary"

Rudimental vehicles will certainly be a thing, powered by steam. Other answers have gone into some detail about this. You can reasonably expect a well-supplied farmer to have a traction engine for ploughing, harvesting and threshing. As with all traction engines, they'll be big, slow, lumbering beasts of burden.

More than that - just no. Even high-pressure steam requires more metallurgical skill than your world possesses, so 1900s steam trucks and cars are a non-starter. Internal combustion engines were only possible once this level of metallurgy was achieved, so the same principle rules them out entirely.

As for motorbikes, have you even stopped to think about how sophisticated an engine needs to be for a motorbike? The precision engineering to make something that small, with that kind of power output? Your world is a century away from that.

The other answers concentrated on the metallurgic aspects of a diesel car, but there are other things missing, specifically I'm thinking of rubber tyres. In the described scenario (essentially isolated villages of 500 people) there is probably a lack of natural rubber that cannot be replaced by something else.

So, no, there will be no diesel vehicles under the given circumstances.