As on so many issues, there is no specific Church teaching on this subject. Still, one can apply basic moral principles (coupled with basic factual information about deer hunting and related issues) to derive an accurate conclusion. While some folks may not like it, the basic answer is that deer hunting for sport is not morally problematic.
As the Catechism of the Catholic Church indicates, it is wrong to cause animals to suffer needlessly (cf. CCC 2418). This is not because it is a violation of the animal’s rights—animals do not have rights. Instead, the immorality lies in the fact that torturing any living thing fosters a delight in suffering and is thus an abuse of human nature.
It would be wrong to hunt out of a desire to make animals suffer, but in general most hunters seek to minimize the suffering of deer by using shots designed to make the animal’s death as quick and painless as possible. When this fails and an animal is injured but not killed, it is part of the hunter’s code to put it out of its misery as quickly as possible.
Hunting for sport provides a valuable service to nature as well. Wildlife in general—and deer in particular—have a tendency to overpopulate the areas in which they reside. Each area can support only a certain number of animals of a given kind, and the excess animals die of starvation and sickness unless they are culled.
What the state wildlife commissions do is calculate how many deer can be supported in the local environment, study the deer population to see how many have been born in a particular period, and determine the difference. The number of deer licenses that are granted to hunters in a given year is based on the number of deer that the land cannot support and would die anyway.
By allowing deer to be hunted, the deer population suffers less than if it was left to suffer starvation and sickness. Hunters who serve in this culling program are thus playing an important role in natural resource management. States must hire professional hunters to fulfill this role when sport hunters cannot.