Hey there!
Answer:
If you are asking for in the actual games, there are no punishments for hunting. I think the question is what happens if you hunt in the districts. Well, the main character Katniss Everdeen is constantly hunting for food outside the walls of her district which is district 12. In her district, usually they punish people who do this with public whipping but the peace keepers pretend to not notice due to the poor state of starvation the people are in.
I hope that was helpful!
Good Luck :D
(Btw, the Hunger Games are one of my favorite series of books! ;P)