That was my first thought, too Carol. Tie some rope from the base of the plant along the fence. Be sure to loop around a fence post so it will be secure. Train the "arms" of the plant along the ropes. You could flop the top of it over the fence and let your neighbor "share" in the bounty of tomatoes too, if you like.
Tomatoes are naturally sprawling vines and will get as big as you'll let them. It will keep on growing and producing as long as the weather is warm enough, and you keep watering and feeding it.