The Rohirrim explicitly have no issue with female warriors. Eowyn calls herself a "shieldmaiden" like that's an actual thing, and while Theoden and Eomer and Aragorn are all eager to stuff her into a safe hidey-hole the whole war through, it's because they have a different perspective on her duty, not because any of them are under the delusion that she's not capable of fighting. She's clearly well-trained, and since no one acts like that's odd or new news, Theoden would have to have been complicit in said training. And nothing about their relationship says "I'm going to get her raised to the manly arts in defiance of tradition." Rather the opposite, in fact.
So.... where are all the rest of the female Riders?