The "brush killer" you see at most stores (containing dilute triclopyr, and usually some 2,4-d) will likely do the trick.
Usually won't hurt a thick-barked tree at those concentrations, unless you start going crazy and spraying lots of the tree's foliage.
Can also cut the main vines at bottom, and quickly paint the bases with triclopyr/diesel mix (or crop oil for diesel).
Virginia creeper??