I've seen people camp at Cowley State Lake, just east of Arkansas City. Unlighted pit toilets, no hookups, no potable water, so it is camping.
The largest state park in Kansas is at Eldorado Lake (it is a reservoir). The "developed" or "modern" (whatever they call it in the Kansas SP system) camping areas are full hookup, about 1/3 of all designated campsites, 165 of them. The other 300 or so are water/electric. I think the FHU sites may be in the Walnut River area, which is on the floodplain below the dam, not very wooded. The other camping areas are on points of land between coves, east and west sides of the reservoir.
I haven't yet checked out Santa Fe Lake, the closest impoundment to Wichita. My daughter recently moved there, and after checking out what is available in the area we kind of decided I wouldn't take the RV there, rather I'd take it to one of the nicer reservoirs in Oklahoma, and they could drive down from Kansas.
The closest is Kaw Lake, with about a half dozen COE campgrounds currently open. That's accessed going east from Ponca City. Our preference (because it is small and quiet) is Hulah Lake, where the Osage Nation runs a park they took over from the state, but that's about 130-140 miles from Wichita.
Tom Test
Itasca Spirit 29B