Once you pull/cut the vines there's no need to use chemicals on them. Stuff them in black plastic trash bags, seal up and leave them in the sun. Then send those bags to the land fill, NOT to your compost bin. Seeds may still be viable.
Note, though! If you pull or cut or otherwise kill the vines and just leave them laying, they certainly will set seeds and start up again. You do need to remove them completely.Same with after they die back in the fall. Still need to remove as much of the vines as possible.
