Yes and no
Forrest,
The north road from Mammoth to Cooke City remains open year 'round because the other access to Cooke City, Montana (Beartooth Pass) is impossible to keep passable. As I recall, its almost 11,000 at the pass. Anyway, that allows the park service to keep the campground at Mammoth open all year. From what I've been able to find, they close down all of the other roads in the Park to wheeled vehicles about mid-December.
We did the W. Yellowstone/ Snow Coach into Old Faithful last February. It was great - I can't say enough good about the trip. If you like Yellowstone, you'll love it in the winter!
This trip we are planning in late November we are going to try sighting some wolves. We camped at Grand Canyon 2 years back over Thanksgiving and almost had the place to ourselves. I am guessing Yellowstone will be pretty quite that time of year too, but you never know.
PM me if you want some details about our Snow Coach trip. By the way, looks like we are neighbors.
-Vic