I have to tell you that before you run out and buy a goat, They do not all eat poison ivy! The song says "little lambs eat ivy" I wouldn't bet on that either. I have a goat who won't go near it.
Anyway, cut the vine on your tree, about a foot off the ground. You have to take a piece of the vine out or it may just heal back together. Now boil a big pot of water (add a bunch of table salt if nothing good is planted around it) pour it on th leaves and base of the remaining stump of ivy. I would do it every day for week. Once with salt, then just water. The stuff in the tree should die, although sometimes may have shallow roots into the tree and survive for a bit.
If it's on the ground, big vines are more difficult because they lay down roots wherever the plant touches earth. I would try hot water first, but may have to use a brush killer.
Definately don't spray it. Put on a rubber glove and then a cotton work glove and dip your fingertips into the poison. Now give that nasty ivy plant a massage! If putting your hand in it scares you, use a paint brush. Keep your chickens in a run, so they don't eat the leaves.
I have also used black plastic to kill weeds, but it takes a long time and doesn't kill the tough woody stuff like ivy.
I'm in zone 4 NY, might be different where you are.