Epacadostat and Pembrolizumab in Treating Patients With Metastatic or Unresectable Gastroesophageal Junction or Gastric Cancer

INCLUSION CRITERIA * ≥ 18 years of age on day of consent * Histologically-or cytologically-confirmed adenocarcinoma of the distal esophagus \[within 5 centimeters of the gastroesophageal junction (GEJ)\], gastroesophageal junction or stomach, including HER2+ disease * Metastatic or unresectable disease, including those with HER2+ disease * Progressed on at least 1 line of prior therapy for metastatic disease, or intolerant to that therapy if not progressed * If HER2+ disease, should have received prior trastuzumab * Life expectancy ≥ 12 weeks * Eastern Cooperative Oncology (ECOG) Performance Status 0 or 1 * Measurable disease per RECIST v1.1, assessed within 4 weeks prior to study entry * Tumor deemed amenable to biopsy by core for metastatic site or endoscopic biopsy for primary tumor (for both before and on-treatment biopsies) * Able to swallow pills * Female subject of childbearing potential should have a negative urine or serum pregnancy within 3 days prior to receiving the first dose of study medication. If the urine test is positive or cannot be confirmed as negative, a serum pregnancy test will be required * Female subjects of childbearing potential must be willing to use an adequate method of contraception starting with the date of consent through 120 days after the last dose of study medication. Note: Abstinence is acceptable if this is the usual lifestyle and preferred contraception for the subject * Male subjects of childbearing potential must agree to use an adequate method of contraception starting with the date of consent through 120 days after the last dose of study therapy. Note: Abstinence is acceptable if this is the usual lifestyle and preferred contraception for the subject * Prior authorization by Merck in order to enroll in this study is required if previously treated on any Merck-sponsored pembrolizumab-containing gastric cancer pivotal trial * Willing to undergo 2 biopsies (before and on-treatment), provided the procedure is not deemed high-risk and is clinically feasible * Willing and able to provide written informed consent/assent EXCLUSION CRITERIA * Known additional malignancy that has progressed or requires active treatment; exceptions include basal cell carcinoma of the skin, squamous cell carcinoma of the skin, or in situ cervical cancer that has undergone potentially curative therapy * Known active central nervous system (CNS) metastases and/or carcinomatous meningitis. EXCEPTION: subjects with previously-treated brain metastases may participate provided they are stable (without evidence of progression by imaging for at least four weeks prior to the first dose of study treatment and any neurologic symptoms have returned to baseline), have no evidence of new or enlarging brain metastases, and are not using steroids for at least 7 days prior to study treatment; this exception does not include carcinomatous meningitis which is excluded regardless of clinical stability. Patients with prior CNS metastases treated with prior radiation therapy (RT) will also need ALL of the following: * 2 months off RT before starting study or 4 weeks following radiation therapy (XRT) if magnetic resonance imaging (MRI) is stable and the patient is off steroids * Baseline MRI with no edema * Stable for at least 8 weeks * Diagnosis of immunodeficiency or is receiving systemic steroid therapy or any other form of immunosuppressive therapy within 7 days prior to the first dose of study medication * Use of systemic corticosteroids * Currently, or within 4 weeks of the first planned dose of treatment, receiving an investigational agent and using an investigational device * Prior anti-cancer monoclonal antibody (mAb) within 4 weeks prior to study day 1, or anyone has not recovered from adverse events (ie, to baseline or ≤ grade 1) due to agents administered more than 4 weeks earlier (EXCEPTION: denosumab for bone metastases is allowed) * Prior chemotherapy; targeted small molecule therapy; or radiation therapy within 2 weeks prior to study day 1 or who has not recovered from adverse events (ie, to baseline or ≤ grade 1) due to a previously administered agent (EXCEPTION: ≤ grade 2 neuropathy). Recovery from major surgery must be considered adequate prior to starting therapy. * Prior therapy with indoleamine-pyrrole 2,3-dioxygenase (IDO)-inhibitors * Prior therapy with monoamine oxidase inhibitors within 21 days before screening * Presence of a gastrointestinal condition that may affect drug absorption * Active autoimmune disease that has required systemic treatment in the past 2 years (ie, with use of disease-modifying agents; corticosteroids; or immunosuppressive drugs). EXCEPTION: replacement therapy (eg, thyroxine, insulin, or physiologic corticosteroid replacement therapy for adrenal or pituitary insufficiency, etc) is not considered systemic treatment * Known hypersensitivity to pembrolizumab and/or epacadostat or any of their excipients * Known allergy or reaction to any component of either study drug or formulation components * Received a live vaccine, including live attenuated vaccines (eg, Flu-Mist), within 30 days of planned start of study therapy. EXCEPTION: inactivated flu vaccines such as seasonal influenza vaccines for injection are allowed * Known active hepatitis B (eg, hepatitis B surface antigen \[HBsAg\] reactive) * Known active hepatitis C (eg, hepatitis C virus \[HCV\] ribonucleic acid \[RNA\] \[qualitative\] detected) * Known history of active tuberculosis (Bacillus tuberculosis) * Known history of human immunodeficiency virus (HIV) (HIV 1-2 antibodies) * Known history of, or any evidence of active, non-infectious pneumonitis * History of serotonin syndrome after receiving 1 or more serotonergic drugs * Known psychiatric or substance abuse disorders that would interfere with cooperation with the requirements of the study * History or presence of an abnormal electrocardiogram (ECG) which, in the investigator's opinion, is clinically significant * Corrected QT Fredericia's formula (QTcF) ≥ 480 ms or presence of a left bundle branch block (LBBB); if the QRS duration \> 120ms, the JTc can be used in place of the QTcF; the JTc must be \< 340 ms * Active infection requiring systemic therapy * Pregnant or breastfeeding, or expecting to conceive or father children within the projected duration of the study, starting with pre-screening or screening visit through 120 days after the last dose of study treatment * History or current evidence of any condition, therapy, or laboratory abnormality that might confound the results of the study, interfere with the subject's participation for the full duration of the study, or is not in the best interest of the subject to participate, in the opinion of the treating investigator