Éowyn's departure left Rohan without a representative of the royal family but not without leadership.
Per the Tolkien Gateway (citing "Unfinished Tales") Erkenbrand, was left in charge of the Rohirrim defenders, even when Théoden believed that Eowyn would also be there.
Erkenbrand did not come to Minas Tirith to fight in the Battle of the
Pelennor Fields. King Théoden gave him the command of the forces that
were left behind to defend Rohan because of Erkenbrand's dignity and
authority.
Since Éowyn was Théoden's Steward, her absence might harm Rohirrim morale (because of suspicions that she might have fled to some place of safety), and if parley was necessary, the enemy leaders might not consider Erkenbrand an appropriate person to negotiate with, but Éowyn's departure did not leave Rohan without leadership, just without royal family leadership