You will need to use a systemic brush killer. If you are worried about damaging surrounding vegetation, you can paint the brush killer on...with a brush (That always makes me giggle) rather than using a sprayer. You have to get the timing right, like on a hot day with no rain predicted and all that, but it does work. We cleared Poison Ivy from several properties when we worked landscape maintenance.
We had one situation that was particularly tricky. I took the very slow but sure method of partially filling a plastic coffee can with the brush killer, using clothes pins to pin a few of the Poison Ivy vines in the can so they could suck up the chemical. It's a very slow method but you don't harm any other plants. Just make sure the container/coffee can is secured and covered.
