There are USACE recreational access facilities all along the Mississippi, often at the lock and dam sites.
These Corps parks will not be like Tom Sawyer, they are campgrounds rather than RV parks, Several grades of CG from primitive with pit toilets to improved RV sites with power and sometimes water. Only the higher grade CGs will be listed in recreation.gov, and other the sites unconditionally available will be reservable.
Depending on location, COE parls may not be open year round, as they rely on volunteer help which is seasonal. They will also not be available when sites are flooded, though the CGs that don't flood will be what you find available on the resrevations site.
I've stayed in a RV park in Sikeston, nothing like Tom Sawyer but full facilites. Don't remember the name, but it was just east of I-55 on US-60.