But it can be tough to get a place. We got our GT camping reservations several months ago. We like to stay at Bridge bay campground in YNP. I've actually only stayed two nights inside GT when I climbed The Grand. Otherwise, If I'm camping, there are primitive sites in the Forrest and developed sites in the forrest adjacent to GT park.
I've never stayed in West but it's a fun little town. We usually finish with dinner there after a day in YN park then we head home to Jackson.
There are all kinds of places to stay in Jackson, limited places in Gardiner, and West/Island Park. My bet is your more affordable ABNBs are in Island Park.
Alpine, Wyoming and Cody, Wyoming are options but that's an awful lot of driving to get to Yellowstone. Same thing goes for Teton Valley Idaho. Having said that, people do it all the time.

Shoot me a boardmail for more specific questions. I've lived in Jackson for 25 years.