When I lived in Dallas I grew 4 grape vines in the back yard. I had a Lebanese friend who used to come over and pick the young leaves. She used to stuff them with minced lamb, herbs and things. They were OK, nothing spectacular but definitely OK.
If the vines are small enough that you can spray all or most of the leaves without too much collateral damage, I use Round-up or Brush-B-Gone. If they are too big for that, cut them off, drill a couple of holes in the stump (depends on the diameter of the stump), cover the exposed wood and fill the holes with undiluted herbicide. I think just about any herbicide will do, straight 2-4-D is probably the cheapest. Grapes can sometimes be hard to kill because they send out long, long lateral roots. I dug grape vine roots out of my raised beds in Dallas and they were 20 to 25 feet away from the plants.