The National Park Service runs Dunewood Campground, which provides RV (no hookups) and tent camping sites just 1.5 miles south of Lake Michigan.
"The sites are large, private, and separated by groves of trees. All are primitive, but with great access to clean, non-obtrusive restroom/hot shower facilities. Each site has a large round fire pit and a picnic table. It's close to Lake Michigan beaches. The staff was very helpful. Quiet hours of 10-6 were observed by all campers with no problem. It's a very clean campground, with lots of beautiful walking trails that are well marked. There are 25 really beautiful walk-in sites as well." ~ GardenThe State Park campground has campsites with 50 amp electric service and they are within a 1/2 mile walk of the beach. Avid bird lovers will find plenty of viewing opportunities in the marshes that provide haven to over 250 varieties of birds. There are hiking trails too, one of which, the West Beach trail, demonstrates along its course the geological succession from sandy beaches to hardwood forests. Other recommended activities include horse-back riding along Ly-co-ki-we Trail, bicycling along forest trails, hang gliding from the sand dunes at Mt. Baldy, and great Lake Michigan fishing for coho salmon or lake trout. Bikers will find a BMX track and a mountain bike trail at Imagination Glen Park in Portage.
Sounds like a great camping destination, and it's within 2 hours drive from Chicago, IL or South Bend, IN.
Indiana Dunes resources: