I am another who is highly allergic to this demon vine.
There was a patch of it around the base of a tree and after ignoring it for 15 years, it had risen to the top of now dead tree and spread out in a huge patch. I would have continued to ignore it if my cat did not wallow in it and then rub all over my face, neck and chest. I missed two days of work, had two rounds of prednisone and a steroid shot but still was miserable.
I eradicated it by covering my hands and forearms with bread bags and methodically pulling it up over a period of a month or so. In order to keep the vines from swinging around, I would pull a piece of the vine and "fold" it up using my bag covered hand. Repeat until I had a large handful or the bag began slipping down my arm. At this point, I would pull the opening of the bag (by my elbow) away from my arm, encasing the poison ivy in the bag, then throw the lot into a contractor bag that was folded open over a trash can, making sure the bread bags did not come in contact with the top 15-20 inches of the bag. Then I would tie it closed and dispose of it.
I keep bags on hand for the occasional stray plant and have great luck keeping it under control. Spring is the best time to get at it, before the leaves are so large and floppy.