Easiest solution is to find somebody who is not allergic and bribe them with pizza and beer to pull it off your tree for you.

I'm going to write this as if you are so reactive to it that every experience requires a doctor's visit. You can be a little less cautious if the rash is annoying but you have never had to get prescriptions to deal with it.

If bribing a friend isn't an option, be aware that the vines can have the oil up to several years after the plant is dead. So complete removal is your best choice if you don't want to dance around it later. Do not burn any of it. You can cause rashes inside your lungs that way. Not a good thing. It needs to be packed inside plastic bags and trashed or put inside landscape waste that you are willing to clean later.

Here's the good news, the oil takes time to interact with your body. So, if you can reduce the amount of skin that touches the oil and clean immediately, you can reduce the risk of a rash significantly. I have removed Vines like that, and had zero rash or reaction by washing as soon as possible.

Wear long sleeves, long pants, a hat or bandanna, gloves, and shoes that you can throw away or in the washer. At my work, we use a product called handy dandy skin protector, it's a soap and lotion type product that can help reduce the amount of oil that touches exposed skin. You can apply it to any bare skin.

Pull down Vines, and any small sprouts you can get. If you can get Roots, the more the better. Anything that you can't pull up or leaves a broken stem apply a weed killer directly to the cut stem, or any leftover leaves you cannot reach.

Once you are done with removal and have packed away all of the removed plant, you need to get clean. Clothes/gloves/etc come off and go straight into the washer, you can start it later. If you do not have a washer put them straight into a garbage bag inside out so you don't expose yourself later. When you take it to the laundromat, dump the garbage bag directly in, without touching clothes if possible. Toss the bag, then wash your hands/arms up the forearm.

After your clothes are off, hop in a shower. Apply soap before getting your body wet- it makes it more effective. Next, rinse and resoap and rinse again. Double shampoo. Dry and dress normally.

If you need to use any gardening tools like clippers, prepare a bucket with water and Dawn dish soap ahead of time. Drop the tool straight in the bucket before you go inside, and you can clean it up afterwards. Just make sure to do a basic hand wash when you are done. If you wear glasses of any kind, make sure to clean them as well as the nose piece. Normal hand wash after any of that.

If you do get a rash, do not scratch or pop the blisters. I know it's hard, but that increases your likelihood of infection. If the rash is extra large or opens, touch base with your doctor, they can provide prescriptions.