During the climbing season of Mr. Fuji, the trails and the huts are quite crowded with plenty of tourists, therefore making a reservation in advance would be strongly recommended.
Especially on Seventh and Eighth stations of Yoshida trail, no sooner has the advance reservations become available than many climbers and travel agencies start to book those huts.
Typically those reservations become available from April 1, so that booking a few weeks or even a few months in advance would be better.
Huts are closed mid September and it is prohibited to climb Mt. Fuji after the trails are closed.
Here is official Web site for Mt. Fuji climbing. http://fujisan-climb.jp/en/index.html
The information is about 2016, but I think 2017 will be published soon and it won’t be big differences.
Huts and trails are closed mid September, so I recommend you to go up at least early September.
Hope this helps you.