Who had a happier childhood in Sohrab in the Kite Runner and Azziza in A Thousand Splendid Suns?

This is an interesting question! Both Sohrab in the Kite Runner and Aziza in A Thousand Splendid Suns did not have what we would consider to be a happy childhood. Both lived in a war-torn country and went through poverty, hunger, and separation from a parent (or parents). The case could be made, however, that overall, Aziza's childhood was happier.
Sohrab's childhood virtually ended after both of his parents were murdered. He ended up in the orphanage, where he was found by Assef, the man who had raped his father when they were both kids. Assef, who was now a Talib, used his power to take Sohrab out of the orphanage and made him his sex slave. Eventually, Amir managed to get Sohrab out, adopt him, and bring him to the United States. However, everything Sohrab experienced resulted in lasting trauma, experiences he is unlikely to ever recover from fully. 
Aziza, on the other hand, was brought up by her loving mother, Laila, and lived with her throughout her childhood. She was also placed in the orphanage at some point, when Rasheed, Laila's husband, claimed he could no longer afford to keep her, but Laila visited her as often as she could manage. Rasheed did not ever act as Aziza's father (partially because she was a girl, and partially because he suspected all along that she was not, in fact, his), so she was not attached to him and his death was not a terrible blow. Eventually, due to Mariam's sacrifice, Aziza and Laila were reunited with Aziza's real father. All things considered, her childhood can be said to have been happier than Sohrab's.