I've never ridden at these places, but I did find some listings near Chicago that do allow snowmobiling.
Kane County lists several forest preserves where snowmobiling is allowed:
http://www.co.kane.il.us/forest/winter.htm#snow
I think that Busse Woods Forest Preserve near Woodfield allows riding:
http://www.trailmonkey.com/ilhike1.htm#Busse Lake and Ned Brown Preserve Trails
Good Luck!