This pilot, randomized phase II trial studies how well depleted immune suppressor stem cell transplant works compared to standard stem cell transplant in enhancing immune response to vaccines in patients with multiple myeloma (MM). Chemotherapy and the patient's own stem cells are effective in treating MM, however there is a risk of disease returning due to poor recovery of the immune system as shown to poor response to vaccines to prevent infections. Before chemotherapy, patients' stem cells are collected and certain immune cells called suppressor cells are removed from the stem cells. Patients then receive chemotherapy to kill cancer cells and after that the immune depleted stem cells are returned to them to replace the blood-forming cells that were destroyed by chemotherapy. Giving depleted immune suppressor stem cells transplant to patients with MM may result in a more robust immune response to vaccines after transplant and may prevent MM from returning. It is not yet known whether depleted immune suppressor stem cell transplant is more effective than standard stem cell transplant in enhancing immune response to vaccines in patients with multiple myeloma.