To add a productive post......
Cut the grape vines in the fall, low and as high as you can reach. Shoot the stump with 50/50 roundup, let what is in the tree be, it will come down.
You have to shoot them in the fall because in the spring and summer they are putting out so much sap you would just be wasting roundup, and winter they are dormant.
Be careful pulling them out of trees. They can be attached to other trees and you can create a real hazard for yourself that a hardhat wont slow down. The other thing it will do is destroy the timber value if you crack the trunk. That will cause a split and two trunks will form. If you are walking in the woods and see a lot of Y trees, that means there were grape vines there.
Sometimes you can kill them by ripping them out, sometimes not. I had a piece at the bottom of a brushpile that was about 12 feet long that started to grow on its own. No root system, the only light it got was filtered by 6-7 feet of brush. There it was, with 1-2 foot tendrils.