My brother bought a property that has several acres of wood. He has a big poison ivy problem that is taking a lot of work to attack. We have been taking down trees and cleaning up brush a few weekends a year for several years. The PI vines are a huge pain in the ass. It is amazing how much foamy sap comes out of those big vines when you sever them.
Pro tip: not first hand experience but second hand - make sure you take off your gloves before taking a piss in the woods.

I might have a suggestion on this. If you can, spray as much of the smaller stuff with a weed killer, including as high up on the vines that go up trees, getting the leaves good helps absorb the spray into the root system. The following year repeat, but it might now not be as bad. Continue until gone. It may take several years for the average homeowner to get rid of it completely.
The larger root vines, going up trees, I try to sever those when I see 'em, right away usually, if I can't see or reach the leaves with spray. If you can reach the leaves, spray, wait a few days, then sever the trunks.
And, of course, if using a chain saw, be very careful to be upwind of the wood chippings and ivy sap. If I'm chainsawing some other wood or trees, and know I've got some poison ivy to cut, I will do that last, or first, it probably doesn't matter, as long as you are careful. Then I feel that when I put my saw away for a couple of weeks or longer, I will clean the saw, again being careful of the residual PI contact. Nasty stuff.