Not only is there no law prohibiting carry while voting, Michigan also does not allow the police, except under very specific situations, to arrest the elector while going to and coming from the polling place or while voting.
MCL600.1825 Arrest; privileged persons.
Sec. 1825. (1) Every elector is privileged from arrest while going to, attending, and returning from elections in all cases except for treason, felony, or breach of the peace.
(2) Senators and representatives are privileged from arrest during sessions of the legislature and for 15 days next before the commencement and after the end of each session.
(3) All officers, warrant officers, and enlisted personnel who are in the actual service of this state or theUnited States are privileged from arrest and imprisonment during the time of their actual service except for treason, felony, or breach of the peace.
There is a belief that this statute applies to "civil arrest" and not criminal arrest. An interesting read on the topic is Michigan Attorney General Opinion 4443 of 1966.
If you read that statute literally, one would think that a soldier or senator can drink and drive and claim immunity. Again, if you read the statute literally, you can beat your spouse and be immune from arrest if you are on your way to vote, voting or returning from voting. That is hardly the case.