I think that the Chicago city marinas have a very definite mooring window. Are you asking about year around live aboard? If so, I think that will be very tough to find.
Try posting to the Lake Michigan email list here on sailnet. Even though it is a very quiet list this time of year, you should get some responses. Good Luck.
By the way, in my opinion, Chicago is one of the best large cities in the US. It lacks very little other than year around sailing. That ice is hard stuff to cut through.