Yes, the grape vines do get heavy with fruit - but if the chickens have access to it, the fruit won't last long at all!
Our vines are on wire - I think it was a medium gauge wire, not super heavy thick, but not thin enough to twist easily. We put turnbuckles between the vines and the posts, so in theory, if the vines stretch out the wire, I can tighten it all back up. Only problem with this theory is that the vines have overtaken the posts, so to access the turnbuckles, the vines need to be bare (i.e. wintertime, and I'm not out there and mindful of doing this in the dead of winter!)
As for the chickens, ours very much enjoy the nice dirt underneath the vines - they spend a lot of time under there digging for pests, taking dust baths, sitting in the shade. Very lovely. However, once the fruit is about 1/2 grown, I restrict their access to this area - I would like to eat the grapes, thank you!