Mackinac Island does not permit the general public to bring motor vehicles to the island. Snowmobiles are common in winter, though.

A few examples of motor vehicles on the island: there are some emergency vehicles; they allowed construction vehicles and a bus to transport workers during road work (the roads are paved); and vintage cars were allowed during filming of the film "Somewhere in Time" (I'm not sure if the production team had motor vehicles at their disposal, though - I seem to recall that the film crews used horse-drawn conveyances, at least for much of their needs).

Also, I'm not sure if they still do this, but the City of Mackinac Island used to keep a police vehicle at the dock in St. Ignace (in the Upper Peninsula) to transport prisoners from the ferry to court on the mainland. Here's the SUV I saw a few years ago: