If it was found out that Katniss illegally hunted, who would punish her, is it the Capitol, or Panem, or someone else?

I don't see an exact reference as to Panem being responsible for punishments, but as the Capitol, we can infer they're in control. None-the-less, Katniss notes poaching carries the severe penalties, and Katniss' possession of a bow would carry the death sentence. If caught, she could be accused of inciting an insurrection.

Even though trespassing in the woods is illegal and poaching carries the severest of penalties, more people would risk it if they had weapons. But most are not bold enough to venture out with just a knife. My bow is a rarity, crafted by my father along with a few others that I keep well hidden in the woods, carefully wrapped in waterproof covers. My father could have made good money selling them, but if the officials found out he would have been publicly executed for inciting a rebellion.