This phase II trial compares the safety and effectiveness of 89Zr-DFO-GmAb positron emission tomography (PET)/computed tomography (CT) compared to contrast-enhanced CT after surgery in detecting clear cell renal cell cancer that has come back (recurrent). For some patients, the risk of recurrence after surgery remains high. Conventional CT methods, such as contrast-enhanced CT, may not detect small volume or micrometastatic disease. PET/CT with radiotracers, such as 89Zr-DFO-GmAb, may improve detection of tumor cells. Girentuximab (GmAb), a monoclonal antibody, is tagged with zirconium-89, a radioactive atom (which is also known as an isotope). The zirconium-89 (89Zr) isotope is attached to girentuximab with desferrioxamine (DFO) and this combined product is called 89Zr-DFO-girentuximab. 89Zr-DFO-girentuximab attaches itself to a protein on the surface of clear cell renal cell tumor cells called CAIX. PET is an established imaging technique that utilizes small amounts of radioactivity attached to very minimal amounts of tracer, in the case of this research, 89Zr-DFO-GmAb. Because some cancers, including clear cell renal cell cancer, take up 89Zr-DFO-GmAb it can be seen with PET. CT utilizes x-rays that traverse body from the outside. CT images provide an exact outline of organs and potential inflammatory tissue where it occurs in patient's body. Using contrast agents with CT scan to enhance the images (contrast-enhanced CT) is standard of care imaging. 89Zr-DFO-GmAb PET/CT may be safe and effective compared to contrast-enhanced CT in detecting recurrent clear cell renal cell cancer after surgery.