This plant is river-bank grap (Vitis riparia). It is a native woody vine found throughout Michigan.
As far as control is concerned, the best course of action for vines usually is to cut them at the base and paint the cut with concentrated glyphosate within 5 minutes of cutting. If the vines are large enough you can also drill a few holes in the stump and gently pour concentrated glyphosate into the holes. An example of such a product is Roundup Weed and Grass Killer Super Concentrate, though there are many other generics available.
When using products containing glyphosate there are a few important points to consider. First, as with any pesticide, remember to read and follow all labeled instructions. Second, glyphosate is a broad-spectrum herbicide, meaning it will injure or kill other plants contacted during application, so care is needed to avoid green plant material, exposed roots, and injured bark of desired plants. Third, glyphosate is relatively safe in the environment when used as labeled. It adsorbs strongly to soil in most cases (i.e. clay and organic matter), allowing even sensitive crops to be planted shortly after application; meaning no carryover issues are expected. Fourth, glyphosate alone can take up to 14 days to show full activity under ideal growing conditions. Retreatment of the area may be needed depending on the degree of infestation. Glyphosate is most effective for perennial control in the fall but can be applied anytime the plants are actively growing (temperatures consistently above 50F). Finally, be sure that the product you choose has only the active ingredient glyphosate or glyphosate + pelargonic acid. Products with additional active ingredients may have other unwanted effects and may delay the planting of other plants in the coming season(s).