Yes, there are mountain huts but reservations are necessary. Please note that the huts provide bare necessities for rest - usually just enough space to lie down - and without much privacy. Since space is limited, the huts are crowded. and men and women stay in the same room in close proximity.
Please see the following page for information on mountain huts for staying on the mountain:
Please also refer to the following guide, which provides comprehensive information about Mt. Fuji: