you have several options. 1) you can call an exterminator/varmint control and have them take care of it. 2) you can get a game cock. 3) you can use a live trap baited with scraps, sardines, tuna, eggs(both cracked and whole, cracked for smell), or some other food. vixen musk (female fox in heat pee) might help. dispose of the animal as you see fit, i suggest a .22 long rifle round to the head. this usually takes a week or more and after setting don't approach the trap again unless it is tripped. 4) the less humane approach. steel traps. WARNING STEEL TRAPS USUALLY BREAK THE LEG OF THE ANIMAL THAT IS CAUGHT!!! WARNING set them around the obvious fox runs, buried about 1/2" to 1" under the surface substance. dirt is best because it doesn't change much. snow melts and exposes the trap, but you work with what you have. set them in sets of 3 or 4 in a rough circle area around bait. follow same procedure as with live trap but understand that if you use steel traps you tend to have to kill what you catch. this could include opossum, raccoon, any of the weasel family(this includes skunks, you can guess how they feel about having something grab them and break their leg.), cats, dogs (these may belong to you or a neighbor, steel traps usually break a leg YOU HAVE BEEN WARNED have fun with the vet bills. large dogs sometimes just have sever bruising), coyotes, and any number of other innocent victims that are not the fox. while most of these are egg stealers or chicken killers remember that Fluffy or Fido can just as easily find the traps. i suggest a .22 rifle for disposal but any gun will work, including shotguns. a shotgun will not damage most steel traps but will destroy a live trap. 5) your last option is to sit and wait on the fox. this is stand hunting. you find a spot where you can set up a blind and wait for the fox. then shoot it when you see it. (good luck with this method.)
now DO NOT USE AN AIR RIFLE!!! you will most likely NOT kill it. instead most likely you will hit it and not do any damage but make it scared or mad. and it either runs off or attacks you. less likely you will hit it and injure it and then you have an injured animal on your hands. it will be hurt, less able to hunt, much more dangerous to your animals because they are easier to hunt, and a danger to you and others in the area, especially children. this is all on top of the animal living in pain. possibly starving to death.
If you insist on "hunting" the fox, use a rifle (center fire cartridge, not rimfire. .22 and .17 are rimfire and are illegal to use while hunting in most areas of the U.S., they can be used to "execute" a trapped animal.) If you are REALLY lucky you have a friend with a pack of beagles of foxhounds and a couple of riding horses. if so you have a very good chance of getting rid of the fox. otherwise i suggest the live trap or game cock methods.
EDIT: adding tips on using live traps.
minimize your scent around the trap. set it and leave it. check it from as far away as possible. don't worry about the bait, if properly baited they won't steal the bait without getting caught. make sure that food is in a position that only can be reached by tripping the trap. place the trap close to the den or the chickens. i suggest the chickens. what ever you catch is probably a danger to your birds anyway. backside of the coop is the best position. the fox is used to you coming close to the coop just don't go too close to the trap. personally i usually have a live trap set close to my coop most of the time. catch them before they become a problem then relocate them.
