I killed one that had been growing under my patio like this:
Take a hammer & screwdriver, chisel or something to split and tear the base deeply. Rip it up & leave ugly, deep cracks. Then pour a thick mixture of compost tea and molasses over it to soak it well. Cover it with mulch and keep it wet. I have killed & rotted out the trumpet vine stump this way, as well as stumps of crape myrtle and holly. Sometimes you have to tear it down deeper to really kill it, but it will work and it's eaiser than digging the darned thing out!
Give it a try...you'll be happy with the results, I'm betting.