Get a couple of bottles of Tecnu and wash your whole body in it beforehand and immediately afterwards.
Good advice. Make sure to be Upwind while pulling the vines from tree.
If I were highly allergic, I would NOT hunt this tree.
As for me, it I have a tree like this, I cut the vines at the base of the tree (in MID-SUMMER), then pull the vines off in late summer. You still need to wash afterward, but the fresh green oil is not as easy to fly all over you.
If things go bad, make sure and post pictures.