If you are going to camp, i.e. tent or bare van, then you will need campground faciliites like toilets, showers, tables, maybe fire rings. If you are going RVing, you might be self contained, but from time to time will want RV park facilities like electrical,water, sewer hookups.
Most public parks with camping have campground facilities, and sections with RV facilities, but some will be primitive on camping, and many the RV accommodation is limited, the amenities you have are those you brought, including power supply.
Most private RV parks will have some sites, usually all, with full RV hookups and extras like pools, recreation rooms, self-service laundries, cable TV, Internet connections, stores and restaurants on site. What they often don't have is a place to camp, many are RV only. Some require you to be self contained in that the do not offer public showers, though will almost always have public toilets if there are any common-use facilities.
So you have to figure out what you are doing, RVing or camping, to figure out where you are going to do it. I know tthat most RVers say they are camping, and call any place the go a campground, but if you don't have a self contained RV, you are really camping and what you need is different from what a RV needs.
That being said, because of the overlap, I take my RV to public campgrounds, including National Parks, state parks, national and state forests, public access facilities on US Army Corps of Engineers water management projects, national and state recreation areas, county parks, and municipal parks. You can RV or camp at almost any of these, but RV facilities may not be as complete as what one expects at RV parks.
But, especially when traveling, I may pull in at night at a RV park, as I would a motel, because of the conveniences offered that I do not expect to find at a campground. Except for some chains like KOA and Jellystone, I do not expect campground facilities at these RV parks, and many do not permit camping, they are RV-only.
National Parks, on a cross country trip, are few and far between, and the most popular are quite crowded in season. What you propose to do, camp out of a car, minivan or van, is certainly feasible. I did it for many years when younger. But when looking for places to stop, you need to be looking for campground in addition to those in national parks, and most of the time RV parks will not work for you because they don't want to see you. When I was camping, most of the places we stopped were fairly primitive, a pit toilet, places to pitch tents and build cooking fires, sometimes tables and water supply, more often not. For hygeine, every few days we would check into a motel, especially when camping off-season when showers and running water at state parks would be closed to prevent freezing.
Memoies of these cross country camping trips was reason my wife insisted, as we retired, that any "camping" we did would be RVing, so that she could bring her house with her and know just what comforts to expect. No more sleeping on frozen ground or cooking in pouring rain, or other surprises like that.
