There is no specific Illinois law that addresses the legality of pistol braces. However, because pistol braces are considered to be a firearm accessory, they would likely be legal to own and possess in Illinois provided you follow all other relevant state laws (e.g., regarding possession of firearms by convicted felons, etc.).
There is no one definitive answer to this question because the law can vary from state to state. In general, though, most people do not consider pistol braces to be firearms and so they are generally legal to possess in most states.
However, it is always advisable to check with an attorney in your state before attempting to possess or use a pistol brace.