The state of Illinois does not have any off-highway, motorized vehicle trails in its state parks system. However, there are private facilities.
This one is just south of Springfield, where I live. It is the South Fork Dirt Riders Club: http://www.southforkdirtriders.com
They are great folks. I would suggest getting in touch with them. If this park is not convenient for you, they may be able to suggest other facilities.
Illinois is hoping to develop more motorized trails in the future. Here is an article I wrote on the subject a few weeks back: http://www.prairiestateoutdoors.com/pso/article/dnr_addressing_concerns_over_ohv_sticker_rules