Just in my neck of the woods in northern Colorado, there are several meet your criteria.
In Estes Park CO, both Paradise on the River and Spruce Lake RV Park have sites that are right along the river at their respective locations.
In the Cache la Poudre Canyon west of Fort Collins, there are at least 6 USFS campgrounds that are right on the river.
In Red Feather Lakes CO, there are 2 USFS campgrounds that are right on their respective lake, Dowdy Lake and West Lake.
That's just right off the top of my head and in my immediate area, within a hour to 90 minute drive from where I live. There are lots of others in Colorado that would work as well.
