Latrelle,
Oh, you have my sympathies! I am terribly allergic to the stuff and I find it completely undesirable. Unfortunately, the only Permies acceptable way of getting rid of the stuff is to pull the vines from the ground. The spreading, lateral vines are typically just beneath the surface and can be pulled with relative ease. Whatever you do with the vines, don’t burn (the smoke will carry the allergen) and don’t let it take
root again (it will form a new plant).
Commonly the lateral vines and climbing vines will sprout from a central taproot. This can be difficult to control. Perhaps use vinegar on this point, or maybe pull as much as possible and then cover with something to prevent re-growth. I don’t often recommend using vinegar, but poison ivy is one of the rare times I will. Also, I only recommend using the vinegar on the parts that can’t be pulled (the tap root part). Spray the foliage and let it die back before pulling the taproot completely. It is stubborn stuff.
If you are pulling by hand, make absolutely certain that you are using gloves and then don’t touch your face or any part of your skin with the gloves. Also (you probably already know this) wear long pants and a long sleeved shirt. Basically cover as much skin as you possibly can, then immediately wash those clothes ASAP.
Unfortunately I can’t think of a plant that controls poison ivy. If you are able to repeatedly mow poison ivy it will have a hard time getting established, but that risks blowing the oil all around while you mow so it is not a great option.
Poison ivy is one of the rare plants that I just don’t like. The stuff grows in full sunlight or the dense shade of a forest where it will climb the other plants to find sunlight. I have seen it vine out over 100’. Pulling is in my mind the best way to control, but it is not a fun task.
I hope this helps,
Eric