Just this summer we shot a fox because it had killed a number of my chickens (atleast 10, and I am still undecided to blame the 13 chicks that i lost this spring to the fox or the coyote family
). I had seen it chasing my birds around the yard, but every time I got out there with a gun, he was gone. But this summer I was tending two chicks that I had in the garage, and my dad came out of the house (the house and garage are attached) and opened the door to the garage ever so slowly. When I looked over at him, he had a gun and was pointing it across the driveway. At this point the chickens weren't even outside yet, so I was wondering what he saw. He aimed and shot. I poked my head out the door to see a nice, fat, chicken fed fox, that had been staring at the shut coop. There was no way that he wasgoing to get into that coop (it is a shed with secure locks and little vents), but I am sure that he could hear them telling me to let them out!
Anyways, all that to say that your neighbors need to stop feeding them and you need to shoot it as soon as possible! They can be tricky buggers (believe me, that one avoided me for 2 years), but eventually, when he doesn't think that you are paying attention, you will get your chance!
Good Luck!