Just wanted to post in this thread because I recently rewatched the entire series and wanted to give my opinions on some of the characters.
Aang - EnFP. A random and off-the-wall kid who has some serious and deep worldviews. However, I will admit that the S/N divide is quite unclear. I think most people want to type him as an ENFP because an NF seems best suited to be the avatar.
Katara - esFJ. I have great difficulty in typing her as well. While she is a definite motherly Fe user, I can't tell much of anything else.
Sokka - EnTp - He is a bit of a chameleon. In the beginning he is a total ESTP, but then his Ne resourcefulness starts to show through as his character grows. Also, he is always the one putting the team on the track and reminding them of their goals, which makes me think Te and just confuses me more. Whatever he is, he's very balanced yet goofy.
Toph - ISTP. Need I say more? Although I will admit her I vs. E distinction is not as clear as the other functions.
Zuko - ISTJ. His obsession with honor screams Si. And although he has feelings, that doesn't make him a feeler! I don't see any Fe either. He is also somewhat similar to Azula, but his energy is often more reflected inward, making him more introverted. Also worth noting, I think his internal struggle has always been due to Si-Fi conflicts. In the end of the series, his Fi grows and is able to work in harmony with his Si. I really love his character development.
Uncle Iroh - INFP. I'm 10000% positive of NFP. He is very happy to go wherever the wind takes him, he always speaking in metaphors, and he always about the complexities of life. I've always wondered what his personality was like before his son's death and if it changed significantly.
Azula - ENTJ. She is extremely efficient, cold, calculating, bossy, and manipulative. Somehow, she always seems to be one step ahead of everyone else. Her mental breakdown at the end was simply brilliant. I applaud the writers for this.
Mai - Some type of boring IxTx.
Appa - INTP