Only you know your traveling habits but if you did a similar trip to Yellowstone in that timeframe then, yes, you can do it.
Keep in mind though that Utah and Arizona will be very HOT in the summer and some of Colorado can be, also.
I'd head to Arizona at the beginning of summer. There are some nice Indian ruins around Flagstaff (and you'll have very cool temps there), do a day trip to Sedona (too hot to stay there) and there are more ruins in that area around Cottonwood. Then do the Grand Canyon and stay IN the park for the best experience. Trailer Village is the place for full hookups. Mather or Desert View have no have hookups.
Then head for the Utah national parks - Zion, Bryce, Capitol Reef, Canyonlands, Arches. For Arches reserve a spot ahead for the Fiery Furnace ranger-led hike. The kids - and you - will love it. You'll be in a shaded canyon with fun walking. All hikes should be done early morning because of the heat. Take lots of water...get a waistpack for everyone to carry extra.
In Bryce do the Wall Street hike. It's amazing! Drive Hwy 12 to get to Capitol Reef - gorgeous.
Don't try to do these parks from one or two base camps. That just involves a lot of unnecessary driving and time. Stay IN the parks.
For Colorado, I'd concentrate on the SW corner - Pagosa Springs (nice soaks and a huge pool of hot spring water), Durango (instead of the costly train ride to Silverton - just stay in Silverton, (the best scenery you'll see on the whole trip) Montrose (Black Canyon Nat'l Park), Cortez for Mesa Verde National Park - take the ranger-led hike to the ruins - involves ladders and crawling through a tunnel - fun.
You'll all be tired after this vacation!
Full-Timed for 16 Years
.... Back in S&B Again
Traveled 8 yr in a 40' 2004 Newmar Dutch Star Motorhome
& 8 yr in a 33' Travel Supreme 5th Wheel