It is not wild grape vine but it seems to be similar so you need to get all the roots so what I may suggest is pruning off the stem close to the ground and then brush on some Roundup to the cut end and see if that works. Otherwise you may in the fall have to remove all of your plants and dig out the whole stem which is some instances may go on for hundreds of feet. You'd be surprised how large the underground stems are. I had grape vine that took over a vacant lot so I just bulldozed all of it.