On the east side, there's several USFS campgrounds that are available to the south of Estes Park, along CO 7/CO 72.
H E R E is a link to some of the sites, specifically Olive Ridge, Camp Dick, and Peaceful Valley. They are all in close proximity to RMNP. They are also quite popular on weekends, so you're going to need to get there early, like shortly after 11am at the latest, when folks are checking out.
On the west side, there's a couple of commercial campgrounds (Winding River) as well as the Timber Creek CG within the park. It's all first come/first served, and though it's popular, it frequently does NOT completely fill up.
In addition, on the west side, there's a variety of USFS CG.
H E R E is the website for those on the west side; they are noted under "Granby" for the town. Every one of them are nice...and popular as well. Some are reservable, some are first come sites.
ETA: Something else to consider, on the east side in Estes Park, two campgrounds that are managed by the Estes Park Recreations and Parks Dept -- Estes Park Campground at Mary's Lake and Estes Park Campground at East Portal. Since you're tenting, East Portal may be more enjoyable, having some basic amenities, but being in the trees. Mary's Lake is nice, but for tents they are in an open, sloping meadow, and there's a residential neighbourhood around the campground. Nevertheless, both might sute your needs.
H E R E is the website for more information on both.