We often stay at KOAs, but they are just one of many options. I check rvparkreviews.com for the area we will stop, find an RV park with good reviews, and call ahead to see if they have a site available.
Unless you will be in a really touristy area you will probably not have a problem finding a site, especially if you call before too late in the day.
Keep in mind that June in Colorado can still be quite cold, and often windy. We have been to south central Colorado the last 2 weeks of June the last 10 yrs. morning temps have generally ranged mid 30's to low 40's, daytime highs 60 to mid 70's.