This phase I trial studies the side effects and best schedule of vaccine therapy with or without sirolimus in treating patients with cancer-testis antigen (NY-ESO-1) expressing solid tumors. Biological therapies, such as sirolimus, may stimulate the immune system in different ways and stop tumor cells from growing. Vaccines made from a person's white blood cells mixed with tumor proteins may help the body build an effective immune response to kill tumor cells that express NY-ESO-1. Infusing the vaccine directly into a lymph node may cause a stronger immune response and kill more tumor cells. It is not yet known whether vaccine therapy works better when given with or without sirolimus in treating solid tumors.