Mike,
You can buy an Illinois non-resident antlerless archery tag over the counter for $15.50 so maybe you can spend a little time on some public ground hunting.
If that doesn't work out, I am about 3 hours north of St. Louis in the SE corner of Iowa. I will be hunting this weekend if you get to bored and up for a drive, shoot me a PM or e-mail, no Iowa tags to be had, but you could sure sit one of my stands and take in the scenary.
Be sure to check out the St. Louis Zoo. Not archery related, but really a nice place. It is open in the winter and free admission. We drive down there a couple times a year.
Chris