Most roads within Yellowstone National Park are not plowed over the winter. From the National Park Service:
Every year in early November, most park roads close to regular traffic as we prepare for the winter season. The only exception is the road between Mammoth Hot Springs and the northeast entrance, which is open to regular traffic all year. Once enough snow accumulates (usually by mid-December), roads open to “oversnow” travel only. This means the only way to visit Old Faithful, the Grand Canyon of the Yellowstone, and other popular destinations during winter is by guided snowmobile or snowcoach, or through our non-commercially guided snowmobile access program. Oversnow travel ends in mid-March, when our plowing crews begin clearing a winter’s worth of snow. Roads start re-opening to normal cars in mid-April.
The National Park service also provides a list of concessionaires who operate winter tours inside the park, either in snowcoaches or on snowmobiles. A few of these are listed as operating from the South Entrance of the park (near Grand Teton); this would be easier than driving all the way to Mammoth. (Moreover, it appears that very few snow tour operators are based in Mammoth anyhow.) As of this writing, the cost for a winter day trip from either Jackson or Flagg Ranch (in Grand Tetons) to Old Faithful appears to be in the range of USD 300–350 per passenger.
There is also a non-commerical snowmobile access program, whereby groups of people can apply for permits to bring snowmobiles into the park:
Permits are awarded through a lottery system hosted by www.recreation.gov (877-444-6777). Permits cost $40/day plus a $6 application fee. Lottery applications are accepted August 1 to 31. Permits are awarded in mid-September.
Unclaimed or canceled permits are made available on a first-come, first-served basis beginning October 9. There is no waiting list. Cancellations may occur throughout the winter season, so check the website often for openings.
In principle, one could rent snowmobiles for this purpose from a local outfitter. However, the deadline for applications for this winter has already passed. It's possible that you could snag a canceled/unclaimed permit, though; if you and your group already have some snowmobile experience, this might be a good option.
Note that the winter "oversnow" season usually begins around December 15th. If you are traveling in early December, you may not be able to reach Old Faithful at all.