I'm afraid I have no experience in that part of the country. However, we have traveled the midwest. We, too, have a travel trailer and 3 children (now teenagers).
I can tell you that there is a place in Nebraska, Eugene T Mooney State Park, that has a stable and daily horserides, as well as horse tack lessons. They also have a water park ON SITE, fishing ponds, paddle boats, a facility with a climbing wall, tunnels and balls (the kind little kids love--and it wears them out), crafts daily, a theater with audience participation (we went to 2 different ones in the week we were there). There is a lot I haven't mentioned, that I'm sure my kids would remind me of. They have a ice cream place. It's midway between Omaha and Lincoln Nebraska. We could have easily stayed 2 weeks, with all there is to do in Omaha (30 minutes away) and Lincoln (30 minutes away), as well as Council Bluffs, which is just over the state line. There's a drive through animal park there, too.
There is also a place in the south part of Missouri, but the drive there is too hard on a person's stomach.
Other than those, most campgrounds we've been to have horses at neighboring farms, for use most any time.