Valid question. In my experience an animal that has lived and breathed Mother Nature's best clean air available is nothing to worry about. You should feel more confident eating something you shot than something you bought. No hormones, residual antibiotics or medications. Don't fool yourself, the animals won't willingly eat stuff that tastes bad.
But, some wild cervids have Chronic Wasting Disease and the general advice is to avoid contaminating the meat with cerebral spinal fluid. That is simple with deer - leave the head and spinal column intact (with the hide attached) until the last step. Hang the deer or elk by its back tendons, and butcher first. The appropriate Wildlife Dept can advise if there are risky areas.
Some moose get a brain worm from deer (or is it the other way around?). It is a parasite that makes them stupid. Same precautions.
The other one to watch for is a bear that has been eating at the garbage dump versus one that is feeding in the wild. Don't hope to get good cuts off a bear you shoot anywhere close to a dump.
Bottom line - if it looks diseased, don't shoot it. The coyotes and bears need to eat for their strength, and they don't have the interwebs for advice.