I can't find any bus service between Jackson and West Yellowstone.
However a different possibility. You may be able to go via snow coach through Yellowstone. Not a regular service. At best two seperate legs/reservations changing at Old Faithful. You might not be able to make the connection and have to plan on stayingat,having a reservation at the snowlodge.
Would be more expensive than a hire car in any case.
Try calling or emailing the snowcoach providers. I know people come in from Jackson and stay in the park, but more come in from West Yellowstone and Gardnier.
The Jackson/Jackson Hole tourist bureau may be able to help.
Staying overnight at Old Faithful in winter would be a memorable part of your vacation.
Snowcoach operators in Yellowstone.