I would spray with something like round-up (the tree will be fine its bark will protect it). Then like a week later (to let the round-up translocate to the roots) I would prune a 4" section above the roots out of the vine, so it can't send any nutrients back and forth. That should at least be a good start.
I was pretty confident that it was poison ivy, until my husband said "no, I get this plant mixed up with poison ivy all the time." Guess I should have accounted for him being a man, and thus being compelled to "make believe." lol
About 3 weeks ago Daniel posted this link, that I hadn't seen before. It strikes me as a very clear one for identifying poison ivy. http://grandpacliff.com/Plants/PoisonIvy.htm
I just went outside with a big trash can and some gloves. I ripped it all down from the tree, and found the big underground root where it had respawned from. I pulled up as much of the big root as I could, and I'll continue to keep an eye on the area. I don't think they can, but I think the only thing that may make it unmanageable is if they respawn from air roots