How do I get rid of weed vines without using lots of chemicals?

If you want to go the absolutely no chemical route you need to dig up as much of the roots as you can. You won’t get them all, but pull and dig up what does resprout. An effective chemical method, using very little herbicide, would be to cut down the vines and dab glyphosate, or brush killer, on the cut ends of the vine. The herbicide will be pulled into the roots where it will kill the plant. There will be some resprouting in either case so you will need to do it again. Eventually the vine will run out of energy and die.

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