Grapes have a deep taproot system. It will be most difficult to get enough removed to insure that they will not re-sprout. Control of the sprouting by purely mechanical means will not be easy, but simply cutting them with a shovel below the ground line each time they come up will eventually exhaust the food reserves in the root system. Another way to kill the vine is to spray it with 2,4-D herbicide before the leaves have been killed by frost. Grapes are highly susceptible to 2,4-D herbicide, and treatment in the fall when the vines are storing food reserves in the root system has an excellent likelihood of killing the entire plant (root system and all). However, be extremely cautious with this herbicide as it can damage nearby plants that may not be an intended target. You may already have had a killing frost in your location or you may have insufficient time from application before frost kills the leaves for movement of the herbicide into the root system. In that event, you may simply have to plan to treat (or cut off) the grape suckers next season as they come up. As with all lawn and garden chemicals, consult and obey the product label recommendations regarding application.
