First, if you do not have a car then McCormick Place may work. I would call them and ask about “event hours”. You really do not want to drive you RV around Chicago to see the sights. Rent a car and enjoy all the attractions mentioned above along the lake front.
There is one very good campground northwest of the city. Paul Wolff Campground is located just northwest of Elgin, IL. Again, rent a car so you can get to the commuter train stop in Elgin or just drive into the city on I90. To get to the lake front takes about an hour (double that time in rush hour).
http://kaneforest.com/ForestPreserveView.aspx?ID=11
I used this campground many times when my son lived close to Elgin. If not being able to reserve a site is a put off. Know this, even on weekends it has never been full. Only once did I have to stay in the new section without trees.