The huts on at the Fifth station open throughout the year and the Sixth station open until early October. However, upper huts from Seventh to the summit unfortunately close on around September 10 as all the trails including public toilets from Fifth station to the summit will be closed at that time.
Climbing season of Mt. Fuji is usually from July 1 to early September. Although still some people climb to the top in offseason, official Mt. Fuji association strongly recommend to go climbing during the climbing season in order to avoid any dangers. You might already know but that the weather on mountains is sometimes unpredictable and it might cause serious accidents.
Hope you find an ideal plan to enjoy climbing and have a special memory in Japan.