I've had snow, enough to cover and close roads, in July, August, September, and October in the Greater Yellowstone area and park. Have a flexible schedule, if the roads close, you'll have to wait it out for a day or so. Pack a few incandescent light bulbs or small heaters to ward off freezing in your compartments, I have a couple of wireless remote thermometers to keep an eye on mine in cold weather. If you stop at a campground with full hook ups, don't leave your water hose connected, just fill your on board tank, drain the hose, and put away until next use.
It's a beautiful time to see the parks, Have a great trip!
Bob & Donna
'98 Gulf Stream Sun Voyager DP being pushed by a '00 Beetle TDI