It's been over 10 years since I've been snowmobiling in Yellowstone, and the rules have become more and more restrictive seemingly every year. That said, there are more miles of trails in the West Yellowstone, MT area outside the park than inside.
My experience (been three times) was in the park the "trail" is the same paved road everyone drives their cars, etc on in the summer. Even ten years ago it was forbidden to go off the trail, and if you got caught - it was a major deal.
IIRC entering at West Yellowstone, "making the loop" and returning was right at 140 miles. The trail could be miserably rough, even though it was groomed every night. We saw a ton of wildlife, especially Bison and Elk, and of course Old Faithful as well as the Grand Canyon of Yellowstone. I was advised the best time of year (in terms of fewest visitors and decent snow) was early January. It was cold but not crowded. West Yellowstone is a neat spot. The streets are plowed, but snow covered and people will drive their sleds on the streets to go grab a beer or food, etc.
Some ideas for you:
1.) The West Yellowstone Chamber of Commerce is a fantastic resource. They are very familiar with the park rules, and can provide you with all kinds of worthwhile information and recommendations. If they don't know the answer they would know who to ask...
2.) I think its about 40 miles or so NE of Jackson, WY, but there's a place called Togwotee Mountain Lodge (pronounced Tow-ga-tee). They get a ton of snow, and have snow when it seems no place else does. There's a motel with a restaurant and a bunch of cabins with cooking facilities. In the winter time it's all about snowmobiling, with some groomed trails that they groom (and connect to the Continental Divide trail system). There are also quite a few areas to go off trail or "play". I was there 2001ish and it was incredibly awesome. I wouldn't hesitate to call them and see how receptive they'd be to some guys with Sno-Cats. They also sell gasoline at a small convenience store at the resort (at least they did back then).
I snowmobile a fair amount in Utah and I'm on the same page as Shortbus. If you are on a trail you (and you alone) are responsible for your speed. There is absolutely no guarantee you're not going to meet someone going the other way. Many times I've been snowmobiling and come upon a groomer operated by the state of Utah. It's also not uncommon to find people X-C skiing on the groomed trails, though that's usually fairly close to the trail head.
If the Tucker project is done I'd love to join you...