You are correct, there is a reciprocal agreement between Illinois and Michigan.
The W2s for one employer should be entered as one W2 in TurboTax unless there were more than one W2 that makes up your total wages for the entire year (or other period of employment if less than a full year) for this employer. This is to eliminate doubling your income on the federal return.
When you enter the W2, you will "add another row" to account for each row of state wages and withholding (image attached).
If there are no withholding taxes paid to Michigan there is nothing to report on the Michigan return and it does not need to be filed.
If there is Michigan withholding then file that return to get a full refund of the taxes paid.
If you file the Michigan return by mail, attach a copy of the Illinois. You may owe some money to Illinois since there was Michigan withholding. If both state returns are filed electronically nothing needs to be mailed.
Do not try to use a "credit for taxes paid to another state" on the Illinois resident return, because you aren't paying any tax, you are requesting a full refund.
Which states have reciprocal agreements?
