Can you pay cash today?

Use the self-pay stations either at the campground office or the day entrance. By writing your address on the envelope, the permit will be mailed to you. In the meantime, use the receipt as your parking pass.

If you buy the Michigan State Park pass for your plate, it is $11 and gets you into any State Park in MI free

Not sure, you might not have to pay at all. There should be a worker there at the entrance.