Psshhh she can stay with me! lol only kidding. I'm actually about 4 hours from Yellowstone so not in the immediate area but I did find Happy Tails Pet Resort in Jackson, which would be a bit of a drive but it's beautiful but I don't really know anything about them.
Are you staying in Alpine?? Alpine is TINY! We're talking less than 1,000 people. Here is a vet I found that has a website:
Home they also offer boarding.
Wish I could help more but Wyoming's a pretty sparse place!