At Mt. Fuji, mountain huts normally have some English speaking staff during the climbing season, so you can directly make a phone call for reservations.
If you go on Yoshida Trail, there are four options on the Eighth station as below.
In case you choose Fujinomiya Trail, there is one option on the Eighth station as below.
During the climbing season of Mr. Fuji, those huts are quite crowded, so making a reservation in advance would be required.