http://tchester.org/sgm/lists/coyote_attacks.html
Since you can't discharge a firearm, you're not really going to be able to hunt them. You'd be wasting your money on a crossbow or regular bow - you need a rifle, because you're not likely to get close enough to be within range for an arrow. Traps could work if you get an experienced coyote trapper to set them for you. If you just go out and get the traps yourself, you'll probably fail. Just the tiniest bit of human scent on the trap - while setting it - is enough that they'll go around it.
Alternative - have your neighbors replace their dogs - with BIG dogs. Some breeds of dogs have no problem ripping coyotes to shreds (and have been bred to do that.) Livestock guard dogs generally don't have much of a problem with coyotes. Though, this is probably an inconvenient method of eliminating the coyotes, unless you want very large pet dogs.