Because it's considered unsporting.
Fox hunters don't shoot foxes; they allow foxhounds to chase down the fox and kill it. The fox therefore has a good chance at escaping—at least in theory. It can evade the dogs by going underground or by running into private land whose owner does not permit hunting.
On the other hand, a fox has little chance against a gun. This gives fox hunters a rationalization for letting themselves off the hook. Convinced that what they are doing is fair and "sporting," fox hunters project their own cruelty on to people who shoot foxes for harming chickens, etc.: "See, we're not the criminals! Foxhounds are natural predators of the fox, so what we are doing is just natural! It's the people who shoot foxes that are cruel and unnatural!"