I'm going to try and sell Oklahoma some more here...
Oklahoma IS centrally located in the US. I-40, I-35 and I-44 are three major interstates that run through here. The weather is nice in the summer, but July and August can get pretty hot. June might be a good time. Kids across the country would be out of school and the weather here is Awesome in June. It's 85 to 90 during the day, and about 70 at night. Tornado season is winding down about then too.
There's tons of parks around the city. Lake Thunderbird is about 15 miles SE of OKC. They have primitive sites and sites where you can pull a school bus sized RV in and you'll have 30A electric and water hookups. It was $8 a night last time I was there, and they never came around and collected anyways. There's a beach there where you can swim in the lake and a boat ramp if you bring a boat with you! If you don't have a bathroom in your bus, they have nice bathrooms with showers there. They would be VERY skoolie friendly there as well. They are also pretty tolerant of loud people and beer drinking if that's your thing.
Another neat place is Roman Nose State Park. It's about 80 miles ENE of OKC. It is a beautiful place with lots of parking for buses and RV's. They are skoolie friendly as well. They also have 30A and 50A hookups and water. The sites are $15 a night. There are beautiful mesa's all around with lots of hiking, mountain biking and horse trails. There is also a small lake there where you can fish and swim. Again, if you don't have a bathroom, Roman Nose does!
Red Rock Canyon state park is about 60 miles east of Oklahoma City and is a very cool place. It's a little canyon that's maybe 200 feet deep and several miles long. You can drive down in there and they have lots of RV spaces with hookups, and primitive sites as well. They also have bathrooms with showers although they're kind of creepy. The park is beautiful. There are big red cliffs and lots of trees. There is a big pond that you can fish in and a creek that runs through the whole place. There's tons of hiking trails and you can rock climb as well. You don't need gear, but if you like to repel, there's places for that too. They have a swimming pool and concession stands too.
My favorite place to camp by far, is Palo Duro Canyon. It is in the Texas panhandle about 20 miles south of Amarillo and about 280 miles from OKC. Palo Duro is the second biggest canyon in the country. It's about 1,000 feet deep and 40 miles long. You can see it on Google Earth. Unlike the Grand Canyon you drive down into Palo Duro. They have lots of VERY nice camp sites with all the hookups. It is $15 a night. Their bathrooms and showers are very nice as well. They have miles and miles of incredible hiking and biking trails. If you like rock climbing, there's plenty of that to do too. Wildlife is everywhere. Every time I go, I see deer, wild turkeys, lizards, snakes, big horn sheep, road runners, painted buntings, hawks, coyotes etc. They have a restaurant, gift shop, concession stand and a neat visitors center there too. The views are incredible and there are some really bizarre rock formations there like hoo doo's. They also have a couple of small caves that you can explore without any gear. Palo Duro would be an incredible place and I don't know of anybody that doesn't love it. Most people have never even heard of it. But it is an awesome place. Check out their website and pictures at http://www.palodurocanyon.com
I stayed at El Dorado state park in Kansas a couple years ago. It is 20 or 30 miles NE of Wichita. The place was VERY nice and seemed to be pretty popular. We had a very nice campsite next to the lake and it was about $15 or so a night if I remember correctly. You're pretty exposed to any weather that comes through though.
If nobody wants to pay to camp at a park, my dad has 15 acres about 10 miles east of OKC that he will let us use. There's no hookups or anything, but you would be close to civilization if you had a problem. Or, there's a Wal-Mart down the street from my house and I see RV's there all the time.
Anyways, these are just my suggestions on some nice places.