Regulations on ways to remove groundhogs vary from one state to another, but poisoning is not listed as an option in states like Indiana, Missouri and New York, as noted on the government websites for those states. Poison can be very hard to control and cause numerous issues without even killing the target animal.
There are some situations where poison is permitted, such as to control starlings and blackbirds, but these types of poisons are very specific, as noted by the Missouri Department of Conservation. Some of them also require that a licensed pest control professional apply and spread the poisons. Having a professional apply any types of poisons for pest control can help ensure that the poison does not get in areas where children, household pets or non-nuisance animals do not get into the poison by accident.