The only road that is open in YNP in the winter is from Gardner to Cooke City -- not the right part of the park for you. There are only two hotels open: the Snow Lodge at Old Faithful, and Mamoth Hot Springs Hotel, at Mamoth. From Big Sky, you'd be closest to West Yellowstone, so you could take a snow coach or (possibly) a snowmobile to the Snow Lodge. If a snowmobile option is available, it must be part of a guided trip, as unguided snowmobile use in the park is not allowed. A snowmobile trip is up in the air at the moment because the winter use plan is being contested in court, and there are conflicting court decisions on whether and how many snowmobiles will be allowed in the park this winter.
You should check out the Xanterra website for information on snow coaches and lodging: http://www.travelyellowstone.com/
I've spent a couple of days in winter in Yellowstone, and it is an incredible experience. Just be aware that it takes a lot of time to get into and out of the park -- so don't shortchange your stay. Also, you wont have much mobility once there, as you either must travel with a guided snowmobile trip or on a snow coach. But the snowshoeing and xc skiing is fantastic, and yellowstone in winter is a very rare experience (and likely to become much rarer).