Keith, as you're experiencing grape species (Vitis spp.) can be tough to control by physical means as they are prone to resprout from roots or root segments. In order for it to work, you need to be persistent to delete the root system resources. 

Herbicides can be effective. The best course of action for vines usually is to cut them at the base, drill a few holes in the stump (if it is large enough around) and pour concentrated glyphosate into the holes or paint it on the cut. This needs to be done within a few minutes of making the cut/holes. An example of such a product is Roundup Weed and Grass Killer Super Concentrate, though there are many other generics available. You can find stump treatment instructions on the label (https://www.roundup.com/sites/g/files/oydgjc121/files/asset_images/products/round_up/kill_and_prevent_weeds/pdf/Roundup_Super_Con.pdf) . Be sure to read and follow all labeled instructions.

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 carry over 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). For woody plants, periods of heavy sap flow in the spring should be avoided. 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).

Please let me know if you have any additional questions.