This study will evaluate a novel therapeutic strategy for patients with WAS who undergo haploidentical transplantation using a parental donor. To reduce the risk of transplant-related toxicities, participants will receive a reduced intensity chemotherapy and antibody regimen (conditioning treatment). Participants will then receive an infusion of donor stem cells depleted of certain white blood cells called T- and B-lymphocytes. The stem cell depletion processing will be done through the use of the investigational CliniMACS device. A certain number of T-lymphocytes will be added back to the processed stem cell graft prior to infusion into the recipient.