As much as I love the parks in Utah, it's hot as hell there all summer. I've gone in the spring & fall only, and I had pretty hot weather in May when I wasn't at high altitude such as Bryce Canyon.
Western Washington & Oregon normally has decent weather in the summer and it's rarely hot west of the Cascades. Depending on where you stay, you could visit Olympic NP, the San Juan islands, Victoria B.C. Olympic NP is big and has at least 3 very different parts to visit. In southern Washington and nearby Oregon there's Mt. St. Helens, Mt. Rainier, Mt. Hood, Columbia gorge, and of course, Portland.
The eastern Sierra Nevada has a lot of diverse attractions, though it can get very hot in the valley, too, as you go south. I suspect that Mammoth might have lots of rentals available because it's a ski area in the winter. In the region, there's Mono Lake, Bodie historic site, Devil's Postpile, fascinating ancient Bristlecone Pine forest at 10,000 ft, Alabama Hills, Manzanar historic site, an unusual Obsidium Dome, superb Indian basket collection at the Eastern California Museum, and the Tuolomne Meadows area of Yosemite all fairly easy to reach on day trips.
Black Hills of South Dakota, maybe. There's a lot to see there. Caves, impressive mammoth fossil site, Crazy Horse memorial, historic towns, and Rushmore. Also, it's close enough for daytrips to places near the Black Hills such as the superb Devils Tower and Badlands NP (the latter is hot in the summer, however).
Northern New Mexico, perhaps?