Our favorite time to visit Yellowstone and Grand Teton is end of May to 3rd week of June. It just gets too crowded after that and campsites are harder to find without advance reservations, which we avoid. Also we don't tow a car so we need to be there at a time of year when parking for our 24ft MH is easier to get at the attractions.
We have experienced snow in mid-June on several of our visits to Yellowstone and Grand Teton, but it melted quickly the same day. Last year we drove in the south entrance on June 2nd and were surprised to see Lewis Lake still frozen. Lots of snow still in the woods too. When we drove back down to Grand Teton on the 16th the snow in the trees was all but gone and Lewis lake was ice free. The next day we had snow flurries at Grand Teton. As already suggested above, bring layers to be ready for a temperature range - lows down to the 20's at night and up to 80's during warmer days are possible.
We have never needed bug spray at either Park in June.
2003 Winnebago Minnie 24F - Ford E-450๐