I probably would not feed the vines if they are commercially grown (like from a grocery store) as they are probably treated with pesticides or fertilizer.  But if you had a source that is untreated you could always try it- I think the “vines” would be more like apple twigs and they would chew the “bark” off as opposed to ingesting the whole thing.

Edit- I know some craft things are made out of “grape vine” like wreaths and stuff- if it was from an untreated source they might play/chew with something like that.

I am leary about grapes with the toxicity that has been shown in dogs.  I don’t give my bunnies any as there is too much sugar anyway (although I do give raisins and craisins).  I give grapes to my parrots but never when my dogs are around.

I would stay away from the grape vines (from the fresh fruit) too. Only give a grape as a treat or they can get addicted to them I’ve heard. I also give raisins as treats.

I do not know about the grape vine and would opt instead for apple or willow branches instead.