Answer: What a lovely arbor!
To be honest I am having a hard time thinking of anything that would eat these vines with the possible exception of slugs when they are barely seedlings. If the culprit is slugs you should see slug slime trails around or on the plants because slugs work at night, and the damage should stop when the vines are a bit bigger. In the meantime, hand picking at night, setting beer-baited traps (slug saloons) and possibly using bait according to the label instructions should work.
Another possibility could be a bunny or a woodchuck, in which case it is possible a homemade pepper or pepper/garlic spray would deter the critter, as might one of the commercial repellents such as GetAway. These critters can be very persistent however, and the surest long term solution would be a fence.
Finally, if the damage is slight and the vines are still growing, they may simply outgrow the damage and be just fine -- as you noted, they are very tough plants!
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