Morgan's not an uncommon name anywhere in the country, but it's not specifically tied to any native tribes.
The tribes in Michigan are the Potowatomie and Ojibwe/Chippewa. My husband is actually a member of the Saginaw band of the Chippewa tribe out of Mt. Pleasant. If you think her native heritage comes from Michigan, it's very easy to trace. There are a handful of bands of both tribes in the state and each tribal office holds records back to the early days of the Northwest Territory. None of the information is online, nor will it ever be. But you can take your grandmother's marriage license and death certificate with you to the tribal offices and see if she was a registered member. At the same time, the names of her parents will be on her marriage license and they can check and see if either parent was a registered member of the tribe.
In Michigan, vital records were skimpy between 1870-1890. The only vital records that exist pre-1870 are marriage licenses and church records. If the tribe can't trace her through her marriage license, then you need to work with the local historical society in the area where she grew up to see if they have records on the local tribes that might help you link her/her parent to a specific band of a specific tribe.