Don't forget the VAST trails! Seriously, snowmobiles can be a bit obnoxious at times (noise and exhaust), but they pack down thousands of miles of trail across the state. True, many snowmobile trails can be like moto trails, with no concern for steep climbs and whatnot, but some of them would make awesome fatbike terrain. Probably a good idea to wear some high-vis clothes, because those guys fly sometimes!
I cross-country ski quite a lot in the Stowe area, though never at the resorts. I have only very rarely come across fatbike tracks while I'm out there, and I've never actually seen someone riding one while I'm out. There could be folks riding the VAST trails and their tracks get covered, but I don't know. If I could afford a fatbike, I would probably hang up my XC skis.
There have been a few threads about winter trail etiquette, so keep that in mind before you head out.
In Stowe, some places off the top of my head that would be fatbike friendly without any special permission, and not much snowmobile traffic:
Rec Path - They often keep the lower half plowed for walkers, but the upper half (from 108 up) is often groomed. Grooming is pretty silly though, because it gets so much foot traffic, it's like skiing/riding on cobbles. That's only about 2.5 miles (one way) of flat terrain, but it's pretty.
Smuggler's Notch Rd - Gate at the bottom is closed in winter, so the road becomes a ski trail. Definitely steep towards the top, but could make a fun ride out of it.
Cotton Brook - Snowmobiles use it in winter, but it seems like the only ones that do are on guided tours, so they're in tight groups and moving slow. Once the smoke clears, it's great. I also ski out onto the reservoir quite a bit (doesn't seem like many other folks do), so sometimes there are tracks out there too. Out there, the snow is often quite firm, since it gets a lot of wind compacting and drifting it.
But overall, there doesn't seem to be many people out there on bikes in winter. The fanatical cyclists of summer turn into fanatical skiers. Tough problem to have eh?