I have a row of grape vines and free range chickens. The main vine I have observed is one that is growing up some chain link and is not apart of my main row. What I noticed is that, after I water around the feeder root zone the chickens seem to enjoy the cool dirt in the hot summer. I haven't noticed any damage or wilt to the vine per se, but if they make their dust bowl to deep, they could damage some feeder roots.
Another problem I've noticed is that, the chickens will sometimes roost on my vine. The chickens feet grip the vine fairly tight and rub the bark off the vine and cause some exterior damage to the vine.
As for my Row of grapes, when my fruit ripen and hang low, The chickens will get directly beneath the fruit and vertically jump for the low berries. This wasn't a huge problem because they were only able to get the very low hanging berries and stopped after about 5 or 6 berries(all they could get).
I had to put half-inch hardware cloth around the base of my plants to protect from wild rabbits(Chewed one of my vines right in half). This has also served as an adequate defense against the chickens. They cant peck young buds through the hardware cloth, and it makes it harder for them to try and reach over and peck them. Once your vines get older they cant really do much to them.