wow, I looked up the DNR in Illinois and they really does have crazy laws about vermin like this... that's crazy!  Raccoons here in Ohio are considered dangerous all year because they are so populous and carry rabies so often.  
It says you can try to get a Nuisance Animal Removal Permit, but even then I bet you have to jump through hoops to destroy them after they're trapped.  Good news is, the trapping season will be starting in November, so you CAN destroy them very soon legally, though, I promise not to tell on you if one accidently climbs into a trap sooner.  Not sure how much a hunting/trapping license costs, but I'd say it's well worth the cost... I normally advocate working within the law, but really, who is going to notice if you mistake the actuall end date of the 'season'?
Legal Status
In Illinois, raccoons are protected as a Furbearer. Raccoons in urban areas that cause property damage or present a threat to public health or safety may be removed if an Illinois Department of Natural Resources District Wildlife Biologist issues an animal removal permit.
In rural areas, a hunting or trapping license is needed to harvest a raccoon. In rural areas, there is no limit to the number of raccoons that an individual with a hunting or trapping license may take. The statewide raccoon hunting season is from early November through mid-February, except during the firearm deer season. Raccoons may be trapped from November through January. IDNR biologists monitor the number of raccoons in Illinois to ensure that hunting and trapping do not negatively impact the population. For full hunting and trapping regulations, visit the IDNR Licenses & Hunting website.