A fox or coyote will often catch and throttle a rabbit or chicken, wait a moment or two till it dies, then run off with it. You will rarely see a bunch of blood and guts at a fox or coyote kill of a small animal.
I always encourage anyone to beef-up their coop and run against predators. But, I subscribe to, if you have predators hanging around, kill them. Most are relentless and will never stop trying to get past your defenses. The argument that another predator will take the dead ones place is proof positive there is no danger of hurting the predator population.
Foxes are voracious eaters and are always hungry. And, they are notoriously curious animals. You can use these traits against them. They really are not very difficult to catch. I hate to repeat myself but, there are many very excellent videos on YouTube by professional trappers showing how to trap foxes and just about any other predator you might encounter. Never before has there been more information or equipment for predator control available to the public. Now, it only depends on how much effort you want to devote to the effort.
Yes, another fox or predator may show up. But, the one you kill won't. One less is one less. Mathematics. I promise predator control works. Dead predators kill no chickens! A game cam is a great place to start. Knowing what you are up against is very important.