There is a seed bank in place that is creating new seedlings. If you can catch them while still in the seedling stage, they are relatively easy to pull out. Once they get bigger, it will take a more concentrated effort :-)

Vinegar will do nothing but cause some shriveling of the foliage. It will have no impact on the root system and hollies will regrow from just stubs. When the plants are in full growth mode - like now - cut off the top growth and apply an herbicide that contains triclopyr directly to the cut end of the stem. It may take more than a single application but it IS effective.

will burning stubs with a torch work?

No. You have to kill the root system. Much like vinegar, torching only kills off the foliage or top growth.

I just hate herbicides and try to avoid them.. But there are times they are needed.. Lately here when it rains we get 2-4" at a time so I worry about runoff. Wish my neighbor had delt with the issue on his side before putting up a fence against mine.

As long as the applied substance has time to dry before any rainfall, there is no run-off. And with direct application - as opposed to broadcast spraying - there is really no concern about the product leaching into groundwater, either. In fact in most cases, if the product does get into the soil, it tends to be bound by soil chemistry until it is broken down into harmless chemical elements by the soil microbiology. The drift from spraying is the biggest concern. And over-application :-)