May can have snow.
June is better. Start early in the Black Hills.
Custer State Park has some very nice cabins. They have an excellent website. This is an outstanding state park, close to Mt. Rushmore, etc.
Take 16 west from Custer, SD (or I 90 west from Spearfish, SD) to Buffalo, WY, take 16 over the mountains and up to Cody. Spend a couple of days at least in Cody and see the Buffalo Bill Western Museum, very good, and if the roads are clear, drive the Chief Joseph scenic highway and enter Yellowstone from the NE gate at Cooke City, MT.
There are several hotels and a multitude of cabins in Yellowstone, but make your reservations as early as you can. If you like the log cabin style, Old Faithful is for you with the old lodge, the newer Snow Lodge or cabins.
Or, the Lake hotel, lodge or cabins near Fishing Bridge.
Or you can drive through the park to West Yellowstone and find more reasonable rates at the several motels. This will be the closest you can get to the park but be outside the park.