The roads shouldn’t be a problem in June but keep in mind it can snow in the upper elevations in Yellowstone any month of the year. If it does snow in June it will melt rather quickly.
If it were me I would make Colter Bay a base camp for visiting Grand Tetons National Park and part of the southern portion of Yellowstone. There are two campgrounds called Colter Bay, one is operated by a concessionaire and has full hookups. The other is operated by the NPS and has no hookups.
Then I would move up to Grizzly RV park in West Yellowstone or if you can get reservations, Fishing Bridge RV park within Yellowstone. Sites at Fishing Bridge are full hookup and are truly on top of each other.
If you plan to go this June you better start calling for reservations right now.
Lamar and Hayden Valleys are great places to see wildlife but remember to keep your eyes on the road.