Why wouldn't they allow it? Samurai families trained their girls in the naginata for home defense and kept them racked around the house. So it was no big step to take up kendo. If your family has a really hot warrior to lead the troops, you send her.

In the rebellion against the Meiji reforms, regiments of women fought, usually with naginata and musket, their female officers carrying daisho (mounted, of course). (The History of Japan in Art has the woodcuts of it.).