Where can I book a hut on Mt. Fuji?

I've been searching the internet but cannot seem to find an official website for it.

Hi!

You can check the contact information of huts from URL below.

http://www.fujisan-climb.jp/en/hospitality/mountain_huts_and_toilets.h...
(for example,Yoshida Trail link →「オンライン空室状況ページへ」→ each hut's HP link)
(almost all of the hut's sites are available in japanese)

Above site is useful to know any other information about Mt.Fuji climbing.

Or below sites might be useful.
https://www.fair-fujinomiya.com/mt.fuji/huts-en.htm
https://www.fujimountainguides.com/mountain-hut-reservations.html

Have a good mountaineering :-)