What are some legitimate organizations that could address one or more human rights abuses and/or social issues in The Kite Runner?

There are quite a few organizations applicable to the issues featured in The Kite Runner. There are also many organizations involved specifically in the ongoing war in Afghanistan, which began in 2001.
The International Committee of the Red Cross (IFRC) is the largest volunteer-based humanitarian network. The organization is involved in addressing sexual abuse of children in Afghanistan. According to their site, the risk of sexual and gender-based violence is highest among boys. This is relevant to the sexual assault experienced by Hassan as a child, as well as that experienced by Hassan's young son, Sohrab.
The World Health Organization has a mental health system in Afghanistan, which is relevant to the psychological trauma Sohrab experienced as a result of his sexual abuse. The organization provides daytime treatment facilities as well as both outpatient and inpatient care in general hospitals.
Save the Children has many objectives in Afghanistan. In particular, they protect children from harm and support them in times of crisis. This organization could have helped both Sohrab and his father heal from their physically and psychologically traumatizing experiences of sexual abuse.
Following the 9/11 terrorist attacks in 2001, George Bush doubled participation of the National Peace Corps Association in Afghanistan as part of the War on Terrorism initiative. According to the Peace Corps website, the organization tackles larger societal issues associated with the Taliban. The Peace Corps engages in "peaceful and constructive dialogue" across communities, connecting people to other organizations assisting the Afghan people. In the context of the novel, imagine if Amir had not learned of Sohrab being held by the Taliban or had not been connected with the sources that allowed him to bring Sohrab back to the United States? Sohrab would most likely have continued to be sexually abused by Assef. If the boy was lucky, he would have ended up in an orphanage rather than been taken in by Amir.