This phase III trial compares the effect of the combination of ramucirumab and pembrolizumab versus standard of care chemotherapy for the treatment of non-small cell lung cancer that is stage IV or that has come back after a period of improvement (recurrent). Ramucirumab is a monoclonal antibody that may prevent the growth of new blood vessels that tumors need to grow. Immunotherapy with monoclonal antibodies, such as pembrolizumab, may help the body's immune system attack the cancer, and may interfere with the ability of tumor cells to grow and spread. Chemotherapy drugs work in different ways to stop the growth of tumor cells, either by killing the cells, by stopping them from dividing, or by stopping them from spreading. This trial may help doctors find out if giving ramucirumab with pembrolizumab is more effective at treating patients with stage IV or recurrent non-small cell lung cancer than standard chemotherapy.