Michigan does not recognize non-resident CCW permits, as in a CCW permit from a state you are not a legal resident of. They do not issue non-resident permits either. You would have to get a Michigan CCW permit as a MI resident. Obviously you can't get one as an IL resident. If you are a LEO, you can CCW in all 50 states to include IL. Can you get a CCW resident permit for MI if you aren't? Do this for MI if possible unless you are a LEO in which case it isn't necessary.
check out this website:
www.handgunlaw.us
click on MI and those laws, reciprocity agreements, and other info will pop up. You can click on IL and other states as well. For federal laws, click on "U.S.A." after going to the above website.