RVing Yellowstone is super expensive and extremely difficult in the summer. There's a 14 day maximum stay in the park, even if you change campgrounds. Only one campground (Fishing Bridge) has hookups. To get a shot you basically have to be a ninja as they'll book out in minutes. Reservations open on the 5th of the 13th month before you want to reserve. So if you want to get a spot in Yellowstone for June 2026, you have about a nano second on May 5th 2025 to book one.I love that part of the country. I told the wife the first year when we retire we're taking the RV there when they open Yellowstone up for the spring and staying the whole season. We might have to move from one campground to another, but we'll be in that general area for the summer.
The easier, albeit not cheap, method is to stay outside the park in RV parks. We now go annually in October for a week (the best time to see the park with no crowds, amazing weather, and wildlife everywhere.) we stay at Grizzly RV Park. In June and October, you will want to have full hookups. It can get into the teens at night and snow a few inches here or there.
But if you really want to do a whole summer I'd suggest seeing all of the Northern Rockies. Make sure you're rig is 4 season capable and you can boondock for at least 5-7 days. At that point you can cut your RV Park fees into a fraction. But don't limit yourself to the immediate Yellowstone/Teton area, move around the whole region. Go up to Glacier, over to Hell's Canyon, swing through the Sawtooth's, Crater's of the Moon and back over to the Yellowstone/Teton area for early fall. Arguably the best rafting, hiking, and fly fishing can all be found in the Northern Rockies.