Night 1 -- Highland Ridge Corp of Engineer Campground, just off i-94 near Eau Galle, Wisconsin, just east of Minneapolis. Might be too soon for you (only 316 miles from Schaumberg), but could work if you leave after work.
Night 2 -- Eggerts Landing Corp of Engineer Campground, north of I-94 near Valley City, ND. (Eau Galle to Valley City mileage 380 mi)
Night 3 -- We've stayed at Lake Elmo State Park, just north of Billings, Montana. Also at Billings KOA. (Eau Galle to Billings mileage 550)
Night 4 -- into Yellowstone. Technically, you are not supposed to pull a trailer into the NE entrance to Yellowstone, on RT 212. So you have to go 212 to 310 to 72/120 into Cody and into Yellowstone ... just under 200 miles. Or, even longer, go over to Bozeman and drop down 89 into Mammoth Hot Springs. This really depends where you get your Yellowstone camping reservations.
Knowing you are going around a National Holiday ... I'd try to book as many reservations as you could ... even at the Army Corps of Engineers campgrounds (some of which require multi-night stays on holiday weekends, so if you are traveling on a weekend, wouldn't work.)
You can book various national forest and COE campgrounds on <a href="http://www.reserveusa.com/index.jsp%5b/url" target="_blank">]http://www.reserveusa.com/index.jsp
Having said all this ... there is something to be said, particularly when you are on a tight time frame ... to just book a series of KOA's, 500 to 600 miles apart. KOA's are always clean, usually have swimming pools for the kids (as well as your tired body). http://www.koakampgrounds.com/
And they are easily to cancel that a national forest or COE campground reservation. Also usually right off the expressway whereas some of the national forest and COE campgrounds might be 15 miles off the road (or further).