Immunotherapy Combination: Irradiated PD-L1 CAR-NK Cells Plus Pembrolizumab Plus N-803 for Subjects With Recurrent/Metastatic Gastric or Head and Neck Cancer

* INCLUSION CRITERIA:
* Gastric/GEJ cancer Cohort:

  * Participants must have metastatic or unresectable locally advanced Gastric/GEJ cancer that has been histologically confirmed.
  * Participants must have radiologically evaluable disease, with or without measurable lesions by RECISTv1.1. Evaluable but non-measurable disease include small lesions (longest diameter \<10mm or pathological lymph nodes with \>=10 to \<15mm short axis), ascites, pleural or pericardial effusion, inflammatory breast disease, lymphangitic involvement of skin or lung, abdominal masses/abdominal organomegaly identified by physical exam that is not measurable by reproducible imaging techniques.
  * Participants must have received or been ineligible to receive first line systemic chemotherapy for Gastric/GEJ cancer. Participants with HER2 positive disease must have received HER2-targeted therapy.
* Head and neck squamous cell carcinoma Cohort

  * Participants must have metastatic or unresectable locally advanced HNSCC that has been histologically confirmed.
  * Participants must have measurable disease by RECISTv1.1. If the participant is one of the first six participants enrolled on study and is part of the safety-lead in, either measurable or evaluable (e.g. ascites, elevated tumor marker, or lesion visualized on imaging) disease will be permitted.
  * Participants must have received or been ineligible to receive first-line systemic chemotherapy and must have received systemic anti-PD-1 therapy (in the first-line or subsequent-line setting).
* Age \>=18 years. Because no dosing or adverse event data are currently available on the use of this investigation combination therapy in participants \<18 years of age, children are excluded from this study, but will be eligible for future pediatric trials.
* ECOG performance status \<2 (Karnofsky \>60%)
* Participants must have adequate organ and marrow function as defined below:

  * leukocytes \>=3,000/mcL
  * absolute neutrophil count \>=1,500/mcL
  * platelets \>=100,000/mcL
  * total bilirubin within normal institutional limits
  * AST(SGOT)/ALT(SGPT) \<=2.5 X institutional upper limit of normal
  * creatinine Creatinine within 1.5X upper limit of normal institutional limits
* Participants with treated brain metastases are eligible if follow-up brain imaging after central nervous system (CNS)-directed therapy shows no evidence of progression.
* Participants with new or progressive brain metastases (active brain metastases) or leptomeningeal disease are eligible if the treating physician determines that immediate CNS specific treatment is not required and is unlikely to be required during the first 7 weeks of therapy.
* Participants with a history of hepatitis C virus (HCV) infection must have been treated and cured. For participants with HCV infection who are currently on treatment, they are eligible if they have an undetectable HCV viral load.
* Participants on therapeutic anticoagulation with warfarin must have an international normalized ratio (INR) that is within target range for their condition at the time of enrollment.
* The effects of PD-L1 t-haNKs with N-803 and pembrolizumab on the developing human fetus are unknown. For this reason and because these investigational agents teratogenicity is unknown, individuals of child-bearing potential and individuals who can father a child must agree to use adequate contraception (hormonal or barrier method of birth control; abstinence) prior to study entry for the duration of study participation and for at least 4 months after last dose of study drug pembrolizumab.
* Ability of subject to understand and the willingness to sign a written informed consent document.

EXCLUSION CRITERIA:

* Participants who are receiving any other investigational agents or concurrent anticancer treatment. Palliative radiotherapy is allowed.
* Participants with concurrent use of systemic steroids (within 10 days of enrollment), except for physiologic doses of systemic steroid replacement or local (topical, nasal, intraarticular or inhaled) steroid use.
* Participants with active systemic autoimmune disease (e.g., lupus erythematosus, rheumatoid arthritis, Addison s disease, autoimmune disease associated with lymphoma, inflammatory bowel disease). Participants with autoimmune endocrine disorders controlled with medical management (e.g. thyroid disorders, type 1 diabetes, or adrenal insufficiency) will not be excluded
* Participants with a history of grade 3 or higher immune-related adverse events attributed to pembrolizumab or other anti-PD-1, anti-PD-L1, or anti-CTLA-4 therapy. This exclusion does not apply to participants with permanent endocrine insufficiencies (e.g. adrenal insufficiency or hypothyroidism) under satisfactory medical management. Additionally, participants with grade 2 adverse events attributed to these classes of agents will be excluded with the exception of rash, transient hyperthyroidism, transient liver enzyme abnormalities or other transient events that resolved without steroids or immunomodulatory agents.
* HIV or HBV infection due to unknown effect of PD-L1 targeting via a CAR or N-803 in these chronic viral infections.
* Uncontrolled intercurrent illness including, but not limited to, ongoing or active infection, symptomatic congestive heart failure, unstable angina pectoris, uncontrolled serious cardiac arrhythmia, clinically significant coagulopathy or psychiatric illness/social situations that would limit compliance with study requirements.
* Pregnant individuals are excluded from this study because PD-L1 targeting via a CAR has unknown potential for teratogenic or abortifacient effects. Because there is an unknown but potential risk for adverse events in nursing infants secondary to treatment of the mother with PD-L1 targeting via a CAR and N-803, nursing should be discontinued if the mother is treated on this study for the duration of study participation and for at least 4 months after last dose of any study drug.