No, there is no tap water available for guests use at Fujisan Hotel, Taiyokan or at any other mountain hut on Mt. Fuji. The only available water source on Mt. Fuji is rain water which is not sufficient enough to provide hundreds of climbers a day with clean water. Also, there is no soap provided at the toilets, thus it is important to bring with you your own hand sanitizer.
Bottled water however is available for purchase.