I’ve been lurking around here for awhile, but as a Montanan, I felt compelled to reply to this! Even though there are a lot of great vegan friendly places in Montana, I don’t know of many along that route. There are plenty of good grocery stores though, so packing a cooler with your own foods would be a good idea. I’ve spent time in that area, but I don’t live there, so there may also be other places I don’t know about. There’s a Compassionate Cooks podcast episode about Colleen’s visit to Yellowstone that might be a good resource for information too.
For grocery stores, the Good Earth Market in Billings and the Bozeman Co-op are both awesome. I think there’s also a natural foods store in Livingston, but I haven’t been there myself. Most of the regular grocery stores will probably have a good selection of produce and some basic vegan staple items. Bozeman is also the home of "We Can't Say It's Cheese", so it's available in a lot of grocery stores around this area - if you can't get it in your area, it's worth trying out. The Hickory Spread with crackers is a great snack!
In Livingston, the 2nd Street Bistro serves gourmet pizza (and other things) with a focus on local foods. They’re only open for dinner and it’s a good idea to make a reservation there, so you could mention that you’re interested in a vegan meal when you call, just to make sure they’re ready to serve you something delicious.
If you end up travelling down to Red Lodge, Café Regis is awesome for breakfast and lunch. They make a point of saying they’re happy to accommodate special dietary needs and they have several vegan options on their menu.
Chico Hot Springs is a wonderful place to stay – the Paradise Valley is beautiful! Soaking in the hot springs is great too - they've got a poolside bar with live music a few nights a week which is really fun! Unfortunately, Chico's menus are pretty heavy on steaks and that sort of thing – so definitely call ahead and see what they can prepare for you. I’ve gotten “off the menu” meals there before - but it may depend on how busy they are and that sort of thing.
In Yellowstone, you’ll find some basic vegan options (veggie sandwiches and snacks), but this is where having a well stocked cooler would come in handy. Yellowstone (especially this time of year before the crowds arrive) is an incredible place – so enjoy! If you’re travelling in from Gardiner like it sounds like you’ll be doing, stopping for a swim in the Boiling River is really fun! Welcome to Montana and I hope you have a great trip!!