Now I'm not from Michigan and know relatively little about their laws, but this seems to be a general enough topic...
It would seem to me they would have SOME form of lease agreement, if just to limit liability. I mean if it's a voting precinct then people other than church officials have practical control of the property and are inviting scores of people who the church can't filter or control onto their property to vote. What happens if a voter or poll watcher damages something or through neglect or carelessness causes expensive damages to the church in question. It would surprise me if they didn't have SOME form of agreement on those lines....
And if they do, then the elections board, precinct, et al are now leasing the property... But real property law is not something I'm heavily knowledgable on. you would know more then me.