Cut down a trumpet vine close to the ground. A pair of lopping shears or a saw may be necessary for the lower, thicker branches.
Dig the trumpet vine out from the ground the best you can, getting up as much of the roots as possible.
Spray the cut root ends with glyphosate, per label directions. If wanted plants are growing near the trumpet vine, paint, rather than spray, the glyphosate on the cut ends of the roots; this will prevent you from accidentally spraying plants you want to keep. Glyphosate will kill any vegetation it comes in contact with.
Spray or paint any new trumpet vine growth with glyphosate. The herbicide will work its way to the remaining roots to kill them. You can also continue to pull up or cut down new growth until the plant starves.