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Probably Mittens with some hotties in them would be your best bet. Put them on top of your toes as well inside your boots. Amazing how that will keep you warm.
Snowmobiling int he park is very limited relative to what it was 20 years ago. The park allows only a limited number of sleds per day, and all must be guided with approved guide services. Guide services are allocated permits. They will take care of you, have good gear and be very aware of the elements and their patrons. SG, are you going out of West Yellowstone or Gardiner?
Snowmobile rental includes: 600cc snowmobile with electric start and reverse, 1 tank of fuel (from our gas pumps only) per rental day, complete set of our high quality outerwear rental clothing (HJC helmet with full face visor, Polar Tech face mask, mittens, one-piece FXR or Cabela’s suit and the warmest Sorrel Caribou boots).
As Grit dog said, they closed it to the general public and only allow commercial guided tours. I'm sure there is a ban on 2 strokes as well. Several studies indicated tens of thousands of gallons of unburnt 2 stoke oil and fuel was being dumped into the ecosystem each year by the snowmobiles, and the noise was scaring a lot of animals at a time when they were vulnerable. There were multiple lawsuits from conservation groups to snowmobile manufacturers and businesses around the park. I think the end result of a limited number of daily trips through commercial tour operators using their approved equipment is a pretty good way to manage things. At the peak before the limiting regs, there were 750 to 1000 sleds in the park per day on many of the peak winter season days. Of course, you had yahoos chasing wildlife, sledding off the roads in places that were illegal, and general poor behavior from only a few, coupled with the pollution, it's probably a good thing that it's so limited these days.