This study will investigate the safety and effectiveness of a modified stem cell transplant procedure for treating leukocyte adhesion deficiency (LAD). LAD is an inherited blood disorder of leukocytes (infection-fighting white blood cells) that leaves patients vulnerable to life-threatening infections. Transplantation of donated stem cells (cells produced by the bone marrow that mature into blood cells) can improve the immune system of patients with LAD. However, this procedure carries a significant risk of death, particularly in patients with active infection because it requires completely suppressing the immune system with high-dose chemotherapy and radiation. In addition, T-cells (a type of white blood cell) from the donor may cause what is called graft vs. host disease (GvHD), in which the donor cells recognize the patient's cells as foreign and mount an immune response to destroy them. To try to reduce these risks, the donor's T-cells will be removed from the rest of the stem cells to be transplanted.