Someone else can correct me if I'm wrong, or feel free to check yourself...but I'm pretty sure that's all wilderness area up there off the road (Mt. Evans Wilderness), in which case, snowmobiles are not allowed at all.
(If you want to make some snowmobile accessed turns elsewhere, let me know...I wouldn't be opposed to a ride
You really don't need one to get up there most of the time, as they do plow that road in the winter (not first priority, so may be a couple days after a storm & not necessarily all the way through, but far enough to access for skinning/hiking...Bierstadt is popular in the winter too...not good skiing, though)
Snow is often sparse up there...it's all pretty exposed, though there are some stashes.