I would imagine that in popular areas as suggested that backcountry campgrounds are on a permit or reservation system. (At least they typically are in National Parks, probably also are in popular National Forest / BLM areas.) Unfortunately, at a popular time of year like now much of that might get filled up.

Almost all of my hiking in that area in Colorado has been day hiking in Rocky Mountain NP. I can share some suggestions if they end up in RMNP. That said, they also will not be far from Brainard Lake Recreational Area which I have heard has some great hikes.

Around the Tetons I believe there is quite a bit of dispersed camping for free / cheap. Just have to be grizzly mindful and nobody pull a Brian Laundrie.
https://www.fs.usda.gov/r04/bridger-teton/recreation/camping-cabins

Similarly, I can probably suggest a few day hiking trails in Grand Teton. If they want to backpack, backcountry campsites may be more difficult to come by.

Edit: If they get up to Yellowstone, just north of West Yellowstone is a campground named Baker's Hole. It appears they have a number of first come first serve campsites. As of now, I'm car camping there in September unless we can find sites in the park.
https://www.recreation.gov/camping/campgrounds/10231392
Certainly need to be aware of grizzlies in this area. Bear mauled a fishman here in 2021.