My sympathies!! There may well be a vine in your area that is providing berries and therefore seeds to the entire neighborhood and beyond via the birds.

"If I applied a herbicide to a freshly cut stump, can that affect my creeper? Basically, can the roots wind up spreading herbicide into the ground?"

It depends on the herbicide you use, I think. There are some herbicides that remain in the plant residues for a while I believe. I usually use glyphosate ( the ingredient in the original Round-up) in full strength undiluted form as a brush killer. It only needs to be applied in a line on the cut stump just inside the bark where the xylem and phloem are, not even on the center of the stump. The advantage of Glyphosate is that is breaks down relatively quickly (a matter of a few weeks from application) and doesn't transfer from one plant to the other or leave active residues in soil. It needs to directly contact an open cut or a leaf surface to be effective. It may need repeat application the following year; I find that use on buckthorn may need repeating. Some herbicides can leave long-term residues and so would want to be avoided.