Can a hormone be absorbed into the body digestively?
This entirely depends on the kind of hormone. Hormones can be chemically completely different things. For example, estrogen is a steroid hormone and is chemically very stable so that it can pass the stomach without being harmed by the acidic conditions (even without enteric coating. AFAIK enteric coating is only used in emergency contraceptives, not in regular ones. And for the emergency contraceptives it is used in order to reduce side effects such as vomiting). As such, estrogen in your food or drink will have an effect on your body and there are many studies suggesting that this is a reason for reduced fertility in men.

On the other hand, insulin is a peptide, which would be digested and its hormone function completely inactivated when swallowed. 

There is even the notion, that the gas nitric oxide may have a hormone function. In that case, it could reach the blood stream even through inhalation. 

In summary, whether or not a hormone can be active after ingestion entirely depends on the kind of hormone.