but yesterday I noticed a few (hundred) new sprouts coming out of my lawn. Shit is everywhere.
Another vote for Round-up here, but there's specific instructions to follow. And it is a cancer causing substance, so wear gloves, eye protection, and facemask (or even a respirator for extensive use).
Round-up works by penetrating into the plant and roots and killing the plant all the way including the roots, so ideally it shouldn't regrow. Basically, you want to only spray Round-up directly (and up close, targeted spray) on the plant leaves (& stems?) of the weed itself, and not spray it discriminately around, because if its sprayed on other plants/grass, they will die too. You could also hold up something as a backsplash, like a small towel, to help ensure excess spray soaks into the towel and doesn't float down to the ground or other plants.
So it seems the best time to use Round-up is when the plant is small with new sprouts -- ideally it should take a lot less spray to kill a small plant than a large plant, so getting to plant while they are still smaller sprouts can help. But with poison ivy, the root system may be very large, so I don't know how that will work.
If I was using it to kill
hundreds of weeds in my yard or had concern of drinking water or soil contamination, I would try to pull out the old/dead roots of the dead plants afterward to try and dispose of them separately.