West Yellowstone does have an airport. It's open and has commercial service during the tourist season (usually June to September). There used to be an air taxi service but I don't think that's been in business for several years (friend of my brother's worked for them). Pretty sure the current commercial flights are through Delta Connection and connect from...wait for it...Salt Lake City.
There is or used to be an Avis that operates all year to serve the snowmobile and hunting crowd and I'm pretty sure Budget is there in the summer.
And there are shuttles and bus service to Bozeman, and the Bozeman airport as well.
I'll consider myself corrected on Delta flying there. I've spent a ton of time in the SL airport and don't remember ever hearing an an anouncement for a flight to West Yellowstone, but since my time was normally spent away from the terminal where most Sky West flights go from I might not have noticed.
When I searched for "car rental" on Google maps I came up with Budget in West Yellowstone, but trying to search for Avis specifically it came up as well. Google doesn't have Avis tagged as car rental for some reason, but it shows both of them at the airport when I zero in on it.