Pembrolizumab, Capecitabine, and Radiation Therapy in Treating Patients With Mismatch-Repair Deficient and Epstein-Barr Virus Positive Gastric Cancer

Inclusion Criteria: * Written informed consent for the trial * Must have operable gastric adenocarcinoma, T2-T4a, N0-N3, M0 * Staging evaluation within 42 days of enrollment consisting of staging laparoscopy, computed tomography (CT) scan of the abdomen and pelvis, and positron emission tomography (PET) scan will be acquired; also endoscopic ultrasound (EUS) tumor and nodal staging will be required * Mismatch repair deficiency as identified by immunohistochemistry or other institutional standard, or Epstein-Barr virus positivity as determined by in situ hybridization or other institutional standard * Be willing to provide tissue from a newly obtained core or excisional biopsy of a tumor lesion; newly-obtained is defined as a specimen obtained up to 10 weeks (70 days) prior to initiation of treatment on day 1; subjects for whom newly-obtained samples cannot be provided (e.g. inaccessible or subject safety concern) may submit an archived specimen (if available from prior biopsy) only upon agreement from the sponsor * Have a performance status of 0 or 1 on the Eastern Cooperative Oncology Group (ECOG) performance scale * Absolute neutrophil count (ANC) \>= 1,500 /mcL * Platelets \>= 100,000 / mcL * Hemoglobin \>= 7 g/dL or \>= 5.6 mmol/L with transfusion or erythropoietin (EPO) dependency * Serum creatinine =\< 1.5 X upper limit of normal (ULN) OR measured or calculated creatinine clearance (glomerular filtration rate \[GFR\] can also be used in place of creatinine or creatinin…