Hi Lori,

My understanding is that, no, poison oak does not grow in Michigan.

A quick google shows up a few publications saying that it does. I think people are mis-identifying poison ivy as poison oak. Poison ivy leaves can be variable and it can grow as a shrub or a vine, so people might mistake it for a different species. 

We do have poison sumac. Here is a link the UM herbarium information on poison sumac. 

And a link to an MSU publication on poison ivy

Take care,