Upvote:3
Several tour outfits run winter climbs (for example: http://www.fujimountainguides.com/off-season-two-day-mt-fuji-tour.html)
Note from the website that permits are not required and apparently huts are available (at least to the tour operator)
I'd also recommend filing this form with the local police, and emailing their contact information for more advice... http://www.fujimountainguides.com/uploads/1/5/3/1/15315858/climbing_information_form.pdf