I love Yellowstone, we would go this summer if it wasn't for the damn high price of gas. I have camped there in June, July, August, and September. September was the best as there were no crowds. We went once the last week of May, first week of June, and the animals were still down in the valley, there is usually still snow on the ground and on the mountains and it is still cold at night, with lows in the high 20s to low 30s at night, highs in high 50s to low 60s.
I have had it snow on us in June and in September while camping in YS. The nighttime fires sure feel good as do the heaters in the tent and campers. Love cooking breakfast over the fire in the mornings, bacon and eggs just seemed to taste so much better.
We drove into the park one year on Labor Day, it was wall-to-wall vehicles leaving and we were the only car entering campsites everywhere, with no crowds anywhere.