I'm having trouble finding parks and forests that allow backpacking and primitive camping near Chicago, IL.
Are there any good locations near Chicago for backpacking?
For any reasonable definition of "near," no.
However, if you hike into the more remote part of Indiana Dunes State Park and set up a tent, no one will bother you. I've done it with a group of 40. The rangers there are primarily concerned with people being drunk/loud in the designated camping area. This park has the advantage of being accessible by public transit (South Shore Line from Millenium Station).
If you must stay compliant with park rules, your best bet is to drive north into WI. Nicolet National Forest is the closest good place I know, at 4.5 hours away.
If you can tolerate a 7 hour drive, I recommend the Porcupine Mountains near Ontonagon, MI.
We just spent a week on the west coast of Michigan. There was great camping in Manistee, Silverlake, Arcadia, Sleeping Bear Dunes, Traverse City, etc. There is free primitive camping and there is also different degrees of modern camping accommodations. Check out MI Rt 22 which connects all these areas.
It took us about 5 hours to get to Chicago.
if you are open to a 5.5-6 hour drive there is the Shawnee National forest down in Southern Illinois. Currently due to the COvid19 situation all of the public recreational areas are closed/parking....but you can find the backpackers parking lot at garden of the gods and other side parking spots (check river to river trail group on facebook)and do backcountry camping anywhere outside of those state park areas...there is about 120 miles of trail. Make sure that you bring everything you need with you as stopping anywhere in the small towns would be breaking the CDC suggestions at this time.