A couple possibilities across the Straits from Mackinaw City, in St. Ignace: there's a KOA there, not too bad a place, and Straits State Park. We have camped at the former, quite a few years ago, but have only driven through the latter. Some nice shaded spots there, and there's a ferry to the island from St. Ignace. Enjoy! Mackinac Island is one of my all-time favorite places. :)
This one is right near everything. I last stayed there about 5 years ago and the site I was on was narrow when slide outs on both sides of trailer were opened. Mackinaw Mill Creek Randu