I am still unsure of where the OP wants to go.
The west side of Michigan (the state) is the east side of Lake Michigan (the lake).
The west side of Lake Michigan is the east side of Wisconsin (and a bit of Illinois).
So where exactly does the OP want to camp?
We did the trip up the eastern shore of Lake Michigan (west side of the state of Michigan) last June. Nearly all the state parks are great and very scenic. Yes, you will need to buy an out of state permit, but if you are going to camp at more than a couple of state parks, I suggest you buy a season permit. Also, we stayed at the Straits State Park campground when we went to Mackinac Island. Great campground with a view of the big bridge from the campground. Also very near to the ferries that take you to the Island.
Note that the campgrounds along the shore of Lake Michigan get busy, so arrive early, or make reservations. Weekends are especially busy.