Is there ANYTHING beside weed killers that will kill poison vine?

Try pouring undiluted white vinegar on the roots. I have not tried this on poison vine, but I have used it on other unwanted plant growth. It is safe for pets and children.

You could try boiling water. That works on many weeds. Also boiling vinegar.

Cut the vine down as much as you can and spray it down with white vinegar and add some salt to the area,providing there are no other plants around.

3 cups white vinegar

1/2 cup table salt

1 tablespoon liquid detergent or soap (I use Dawn) for stick-to-itiveness.

Directions

1. Mix vinegar and salt until the salt is completely dissolved. Stir in liquid dish soap, and pour into a spray bottle.
2. Spray onto the green growing leaves of the plants.
3. Wait a week, then repeat on any survivors. Best time to apply: during a dry spell.

Repeat as necessary.