Let me recommend a couple websites that I've found invaluable in planning trips like this.
The first is recreation.gov
. This site lists all the Corps of Engineers and National Park Service campgrounds, along with some (but, alas, not all) the National Forest ones. Particularly east of the Rockies, the Corps has built many, many splendid camping facilities, usually near dams, locks or other flood control, irrigation or navigation facilities. Many of these campgrounds take reservations, but all have at least some number of walk-up sites. Most are very well-equipped with hot showers and real restroom facilities. Many are also somewhat RV-oriented, but I've yet to encounter one that didn't have a good tent-pitching site. And many, if not most, have dedicated tent sites at a lower cost.
The other is reserveamerica.com
. This site provides access to a lot of state park campgrounds (I'm not sure if they cover every state park in the country, but they've certainly got most of them). A lot easier than searching for the DNR websites of every state, most of which will then refer you back to reserveamerica.com anyway!
If you are thinking about going from Michigan to Texas via the MOA rally in Des Moines, may I suggest a few places I particularly like along the way? Consider the Corps campgrounds at Grant River (Potosi, Wisconsin--tent sites ten feet from the Mississippi, beneath tall bluffs and near all kinds of curvy "driftless area" roads) and Bull Shoals Lake (northern Arkansas--some of the best twisty roads anywhere, and the Lakeview campground has great facilities, cool afternoon breezes and a fine view of the big lake). Or the NPS campgrounds in the Ozark National Riverways of Missouri (Two Rivers or Alley Spring, in particular, both near spectacular MO Route 19). Two state parks I can recommend are Pikes Peak (near McGregor, IA, atop a 400-foot bluff offering a spectacular view of the Mississippi River valley) and Johnson's Shut-Ins (kinda near Farmington, MO, a great, new campground and a "natural water park" that the river cut out of the granite).
I sometimes make reservations, sometimes just drop in depending on season. Either way, the websites are useful in letting you know where camping is available.
Another important thing to know when camping: some states charge admission fees to their parks (and some set really high fees for out-of-state visitors). Wisconsin is horrible in this respect. IL, IA, MO and AR do not charge entry fees, though they do charge for camping. This is one of the reasons I stay at Pikes Peak, IA, which is right across the river from Wisconsin's Wyalusing State Park--eleven bucks for non-electric camping, no admission or parking fees on the Iowa side; vs thirty-four for a non-electric site and park admission at Wyalusing!