A very cool way to experience one of the native cultures in Panama is by staying overnight with them, for example with the Emberas, or by choosing one of the many small island on Kuna Yala. Another place to go camping, are the National Parks. Some parks are protected areas but offer the possibility of sleeping on tents inside their limits for a small fee.
Panama doesn't have a whole lot of traditional camping grounds, so use your imagination and explore. Camping in Panama is about hope, it's about opportunity, and it's about defining yourself, whatever that is in Spanish, for the many generations to come.