2-3 hours is tough. We just moved to Lenexa from Rolla, MO and we still go back to that area for camping/hiking. If you can stretch to 5 hours you'll have a world of opportunities. One of the best places to go for car camping is Council Bluff Lake. You can go fishing, they have a beach, and there's a 12 mile loop around the lake. It doesn't get nearly as much traffic as the state parks either. There's also the Paddy Creek Wilderness which I've seen equestrians camped at, the Berryman Trail has a nice campground, Powder Mill on the Current River has a campground with a really beautiful spring a short walk away. If you're open to extended your drive time, I can send you a long list of places you might be interested in.
One place that surprised me was Prairie State Park just north of Joplin, MO. Its 2-2.5 hours south of KC and is a real tall grass prairie with wild buffalo and was really interesting to hike. There aren't a lot of trails but there is a campground and it wasn't busy at all when I went there, I think I may have been the only non-employee in the park.