Everyone that keeps poultry should make a wire trap. It's just a small portable cage that you can close the door from afar by pulling a cord. It does double duty as a safe place to put a few birds temporairly for treatment or working on their coop.
Trapping is tedious business. It will take a week or so to get a bird comfortable going into the trap for food.
Do you have a large cardboard box? You might get one for free at an appliance shop.
Prop one end of the box up with a stick, open side down. Tie a long string to that stick. Put food under the box, corn often works well. BOSS isn't bad. You can even toss some corn in the general area to get his attention. Then just walk away. Once the bait is gone, put out more bait. After he learns there is food there hide well away from the box holding the other end of the string. Keep the string tight enough so that when you jerk the stick comes flying out and the box falls. Put bait under the trap and wait. Can be tedious.
It's akward. I wouldn't want to go over and ask the neighbor either. If it was dumped, then it doesn;t belong to anyone and there is nobody to care. If you are it's sole source of support (food) then it sort of belongs to you to do as you see fit.