Illinois does indeed have a raccoon season.  Only coyotes and skunks can be shot year around.  However even they have a trapping season.
You can apply for nuisance animal trapping permit or you can call a commercially licensed nuisance animal trapper.    I am a commercially licensed nuisance animal trapper in the state of Illinois and my permit allows me to trap or shoot problem animals all year.
The downside is you'd have to pay to have a commercial trapper catch the animals.  If you had enough land to hunt, you could probably trade animal removal service for permission to hunt your land, since land access is like gold in IL.