You'd have to make sure it was a poison that wasn't dangerous to Humans. If it is then it's a non-starter.
In addition to that, in what manner does it cause the animal to die? Is it quick? Is it painless? Nobody wants an animal to die in agony.
What happens to the poison after it has done it's work in the animal? Contamination would be a major concern. If the intention is to eat the kill, will it be broken down before leaving our digestive system? Flushing it back out into the environment could still leave it potent enough to cause more unwanted animal deaths.
There's also a question as to what the poison does to the flesh of the animal if it is for consumption. Some forms of animal death result in very poor quality meat.
In terms of hunting, poison doesn't seem to me to be a particularly "sporting" method of take-down. The practice of hunting is to hone skills and refine senses. It's supposed to offer a challenge to the hunter.
Any fool with a bag of rat poison could take down a herd of animals, there's no skill involved in that.
Within the meat industry poison is sometimes used in livestock slaughter. CO2, Ar and N2 gas are used in some circumstances to induce unconsciousness and asphyxiation. I think it's usually used on pigs which lose consciousness only a few seconds after breathing in high concentrations of these gases. There is no distress caused to the animal when gas is used so it is widely seen as a highly humane method of dispatch. They don't know there is anything wrong, or that they are in any danger, they just go to sleep.