SBRT +/- Pembrolizumab in Patients With Local-Regionally Recurrent or Second Primary Head and Neck Carcinoma

Inclusion Criteria: * Pathologically (histologically or cytologically) confirmed diagnosis of locoregional recurrent or any new primary squamous cell carcinoma of the head and neck (including of the nasopharynx or paranasal sinus) that is not amenable to curative resection. * Patients for whom curative resection would be medically contraindicated and/or would impose excessive surgical risk are eligible. * Patients who are medically and surgically resectable but for whom surgery would be associated with undue surgical morbidity are eligible. * For purposes of this protocol, undue surgical morbidity would include total glossectomy; carotid artery resection; laryngectomy or pharyngolaryngectomy; and major ablative resection requiring free flap reconstruction. Patients with primary tumors that can be resected without the forgoing are ineligible. * The principal investigators are available to review these criteria on a case by case basis if helpful to the enrolling institution. A new primary HNSCC is defined where any one of the following criteria are met: * Metachronous invasive SCC developing ≥ 6 months after an index HNSCC, more than 3 cm from the index lesion; * SCC developing in the same region as the index SCC if ≥ 36 months after the index diagnosis and if within 3 cm of a site where disease was completely resected or complete response was documented; * New SCC that is cytologically or molecularly distinct from index SCC (e.g. new HPV negative SCC with prior index SCC that was HPV positive). * Tumor tissue testing for p16 status is required for base of tongue, soft palate, and tonsil cancer. If a p16 testing has been previously performed on an oropharynx cancer that has recurred, then repeat testing for p16 status is not required. Participants whose first cancer was an unknown primary must have p16 testing from either the new primary tumor or the recurrent cancer. * Prior radiotherapy (RT) to the head and neck (30 Gy minimum) * Disease must be limited to a single site or adjacent sites that can be treated in a single contiguous target volume for which the maximum total tumor dimension (GTV) must be \<7.5cm. Examples of eligible patients include: 1. A primary site recurrence in the oropharynx with a concurrent level 2 nodal mass, or a laryngeal recurrence with a level 3 nodal mass 2. Multiple nodes in the same (level 2) or adjacent nodal levels (levels 2 and 3) 3. Skull base recurrence with a lateral pharyngeal or high level 2 node Note: These cases will be eligible provided that the maximum total tumor dimension is \<7.5cm. For cases in which a tumor biopsy was performed and there is a biopsy/tumor debulking bed adjacent to the gross residual disease, all of the preoperative radiographic abnormalities must be included in the GTV and meet the \<7.5cm maximal dimension criteria to meet eligibility. Note: Patients who meet these criteria only after surgical removal of a portion of the patient's disease (e.g. removal of level 4 nodal mass in a patient with a tongue base primary; or of a contralateral nodal mass in an N2c patient) are ineligible. * Patients who have undergone a recent biopsy (e.g. incisional) are eligible. Any preceding surgical procedure beyond a biopsy (e.g. debulking) must be reviewed as follows: * Patients rendered free of gross disease are not eligible. * Patients with gross residual disease postoperatively, must be reviewed by the Surgical Co-PI for determination of eligibility. * Patients eligible for study must have cutaneous wounds healed for 4-6 weeks prior to the initiation of SBRT. * History/physical examination within 56 days prior to entry * Examination by a Radiation Oncologist and Medical Oncologist within 56 days prior to entry; \[Note: Baseline dental assessment is strongly recommended prior to start of therapy but is not required for eligibility\] * Contrast enhanced CT or MRI, of the tumor and neck within 56 days prior to entry. * Chest CT scan or full body PET/CT within 56 days prior to entry; patients with equivocal pulmonary nodules that are \< 1.5 cm, that cannot be safely biopsied, or that are negative on PET/CT imaging are eligible. * Zubrod Performance Status of 0-1 within 28 days prior to entry. * Age ≥ 18 * Trial is open to all genders * Hematologic: Absolute neutrophil count (ANC) ≥ 1,500 cells/mm3, Platelets ≥ 100,000 cells/mm3, Hemoglobin ≥ 9 g/dL * Hepatic: Total bilirubin ≤ 1.5 x ULN OR Direct bilirubin ≤ ULN for subjects with total bilirubin levels \> 1.5 x ULN, AST (SGOT) and ALT (SGPT) ≤ 2.5 x ULN * Creatinine ≤ 1.5 x ULN, OR measured or calculated creatinine clearance \> 60 mL/min for subject with creatinine levels \> 1.5 x institutional ULN \[NOTE: Calculated creatinine per institutional standard; GFR may be used in place of creatinine or CrCl\] * Coagulation: International normalized ratio (INR), OR prothrombin time (PT), and Activated partial thromboplastin time (aPTT): ≤1.5 × ULN unless participant is receiving anticoagulant therapy as long as PT or aPTT is within therapeutic range of intended use of anticoagulants * Negative serum pregnancy test within 14 days prior to entry for women of childbearing potential. (Note: A pregnancy test must be repeated within 3 days prior to the administration of the first dose of pembrolizumab) * The patient or legally authorized representative must provide study-specific informed consent prior to study entry. Exclusion Criteria: * Distant metastases. * Tumors that involve more than 180 degrees of the carotid artery that directly communicates with the skin or mucosal surface (e.g., if the tumor communicates with either external air outside of the skin, or internal air inside mucosal surfaces) on diagnostic CT or MRI of the neck within 56 days prior to entry. Note: Tumors that involve \>180 degrees of the carotid, and even those that fully encase the carotid artery are eligible if they do not meet the above criterion. Investigators should email the Study Principal Investigators to review any cases whose eligibility due to carotid encasement are unclear. Investigators are encouraged to review the CT simulation imaging and ensure that tumor progression has not occurred whereby patients who were initially eligible based on diagnostic imaging, would be rendered ineligible based on CT simulation imaging (e.g. tumor size \>7.5cm, skin involvement, \>180 degrees of carotid encasement by tumor). If this does occur, the patient should be removed from the study and the Radiation Oncology Co-PIs should be notified via email. Note: It is strongly recommended that CT simulation be performed prior to entry. * Patients with gross skin involvement (i.e. tumor ulceration through the skin) are excluded. Patients with tumor approaching the skin but in which the overlying skin remains intact are eligible, providing that planning constraints can be achieved without the use of bolus. * Disease that requires two or more discontiguous target volumes will be ineligible. Examples of such cases include: * Bilateral nodal targets * Level 2 and level 4 nodes * An oropharyngeal recurrence with a low level 4 node; * Patients for whom the maximal total tumor dimension (GTV) is \>7.5cm * Prior radiation to primary tumor within 6 months of entry * Prior systemic therapy, investigational agent or investigational device within 28 days of start of study treatment. * Surgical resection of the qualifying cancer is not permitted. (Patients who have undergone biopsies are eligible). Patients without radiographically apparent gross tumor are ineligible. For cases where an operation more extensive than a biopsy was performed but radiographically apparent gross residual tumor remains, will be reviewed by the Surgical Co-PI for determination of eligibility. * No concurrent treatment with other investigational agent or investigational device. * Prior therapy with a checkpoint inhibitor (eg anti-CTLA-4, anti-PD-1 or anti-PD-L1 therapy). * Patients with immunodeficiency, or receiving systemic steroid, or any form of immunosuppressive therapy at the time of registration (e.g. history of human immunodeficiency virus - HIV). Use of physiologic doses corticosteroids may be approved with consultation with study chairs. * Active autoimmune disease that has required systemic treatment in the past 2 years (i.e. with use of disease modifying agents, corticosteroids or immunosuppressive drugs). Replacement therapy (e.g. thyroxin, insulin, or physiologic corticosteroid replacement therapy for adrenal or pituitary insufficiency etc.) is not considered a form of systemic therapy * Known active hepatitis B (positive test for virus surface antigen - HBsAg) or hepatitis C virus (e.g. positive HCV RNA qualitative test). * History of (non-infectious) pneumonitis that required steroids or current pneumonitis. * Treatment with a live vaccine within 30 days of entry. Note: Seasonal influenza vaccines for injection are generally inactivated flu vaccines and are allowed; however intranasal influenza vaccines (e.g., Flu-Mist®) are live attenuated vaccines and are not allowed * Unstable angina and or congestive heart failure requiring hospitalization in the last 6 months. * Transmural myocardial infarction within the last 6 months. * Active bacterial or fungal infection requiring intravenous antibiotic at the time of registration; Note: If the infection resolves and the patient is on p.o. and still within, the required registration timeframe, then the patient is eligible * Chronic Obstructive Pulmonary Disease exacerbation or other respiratory illness requiring hospitalization or precluding study therapy within 30 days of entry. * Other significant medical, surgical or psychiatric conditions or requirements for any medication or treatment that in the opinion of the investigator may interfere with compliance, make administration of anti-PD-L1 therapy hazardous, or obscure interpretation of adverse events (AEs), such as a condition associated with frequent diarrhea. * Pregnancy, nursing females, or women of childbearing potential and men who are sexually active and not willing/able to use medically acceptable forms of contraception; this exclusion is necessary because the treatment involved in this study may be significantly teratogenic.