Early June you should have a good time. There will be snow in a few spots still. The animals are still down in the Hayden valley, but will leave soon. A car for travel would help, and more convenient. The reservation system might not be convenient, call the camp ground @fishing bridge direct. They know what they have better than the reservation system. Just saying, It worked for me. If you stay out of the park on the west side in West Yellowstone, there is a nat'l forest camp ground three miles north on 191(or so)that has water and power, no dump station, Its Baker camp ground. You can use the park dump stations to dump your holding tanks. There is few in there.
Enjoy.