Pembrolizumab With or Without Vismodegib in Treating Metastatic or Unresectable Basal Cell Skin Cancer

INCLUSION CRITERIA * Histologically-proven basal cell carcinoma (BCC) in which curative resection is unlikely without significant morbidity, or have nodal or distantly metastatic disease which has progressed on vismodegib (Arm 1) or has achieved partial response or stable disease on smoothened inhibitor (Arm 2). Individuals who are intolerant or have a medical contra-indication to smoothened inhibitor will be enrolled into Arm 1. * Have measurable disease based on Response Evaluation Criteria In Solid Tumors (RECIST) version 1.1 * ≥ 18 years of age on day of consent. * Willing to provide tissue from a core or excisional biopsy of a tumor lesion up to 6 weeks (42 days) prior to initiation of treatment on Day 1, or availability of adequate archival specimens. * Eastern Cooperative Oncology Group (ECOG) Performance Status = 0 or 1 * Absolute neutrophil count (ANC) ≥ 1,500/mcL * Platelets ≥ 100,000/mcL * Hemoglobin ≥ 9 g/dL, or ≥ 5.6 mmol/L, without transfusion or EPO dependency (within 7 days of assessment) * Serum creatinine ≤ 1.5 x upper limit of normal (ULN) OR measured or calculated creatinine clearance ≥ 60 mL/min for subject(s) with creatinine levels \> 1.5 x ULN * Serum total bilirubin ≤ 1.5 x ULN OR direct bilirubin ≤ ULN for subjects with total bilirubin levels \> 1.5 x ULN * Aspartate aminotransferase (AST) ≤ 2.5 x ULN, OR ≤ 5 x ULN for subjects with liver metastases * alanine aminotransferase (ALT) ≤ 2.5 x ULN, OR ≤ 5 x ULN for subjects with liver metastases * Albumin ≥ 2.5 mg/dL * International normalized ratio (INR) or prothrombin time (PT) ≤ 1.5 x ULN unless subject is receiving anticoagulant therapy, in which case PT/INR must be within therapeutic range of intended use of anticoagulants * Activated partial thromboplastin time (aPTT) ≤ 1.5 x ULN unless subject is receiving anticoagulant therapy, in which case PTT must be within therapeutic range of intended use of anticoagulants * Female subject of childbearing potential should have a negative urine or serum pregnancy within 72 hours 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 should be willing to use 2 methods of birth control or be surgically sterile, or be willing to abstain from heterosexual activity for the course of the study through 120 days after the last dose of study medication. Subjects of childbearing potential are those who have not been surgically sterilized or have not been free from menses for \> 1 year. * If female, is postmenopausal; is surgically sterile; or agrees to use 2 acceptable methods of birth control throughout the trial, until 120 days after the last dose of treatment * If female, agree to use an adequate method of contraception starting with the first dose of study therapy through 120 days after the last dose of study therapy * Male with female partner of childbearing potential agrees to use adequate method of contraception throughout study, until 120 days after last dose of treatment or last blood draw. * Willing and able to provide written informed consent/assent for the trial. Consent may be obtained by legally-authorized representative (LAR). EXCLUSION CRITERIA * Currently receiving investigational study therapy, or has received investigational study therapy, or used an investigational device, within 4 weeks of the first dose of treatment. * 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 trial treatment (EXCEPTIONS: topical, intraarticular, intralesional, and inhaled steroids) * Known history of active Bacillus Tuberculosis (TB) infection * Hypersensitivity to pembrolizumab or any of its excipients. * Has received an anti-cancer monoclonal antibody (mAb) within 4 weeks prior to study Day 1, or who has not recovered (ie, ≤ Grade 1 or at baseline) from adverse events due to agents administered more than 4 weeks earlier. * Has received chemotherapy, targeted small molecule therapy, or radiation therapy within 2 weeks prior to study Day 1, or who has not recovered (ie, ≤ Grade 1 or at baseline) from adverse events due to a previously administered agent. (EXCEPTION: Subjects with ≤ Grade 2 neuropathy may qualify for the study). * If subject received major surgery, they must have recovered adequately from the toxicity and/or complications from the intervention prior to starting therapy. * Known additional malignancy that is progressing or requires active treatment. (EXCEPTIONS: basal cell carcinoma of the skin or squamous cell carcinoma of the skin that has undergone potentially curative therapy, or in situ cervical cancer) * Carcinomatous meningitis without consideration of clinical stability * Known active central nervous system (CNS) metastases and/or carcinomatous meningitis. 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 trial 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 trial treatment. * Active autoimmune disease that has required systemic treatment in the past 2 years (ie, with use of disease modifying agents, corticosteroids or immunosuppressive drugs). Replacement therapy (eg, thyroxine, insulin, or physiologic corticosteroid replacement therapy for adrenal or pituitary insufficiency, etc.) is not considered a form of systemic treatment. Intraarticular, inhaled, and intralesional doses of steroids are allowed at screening and during the study. * Known history of, or any evidence of active, non-infectious pneumonitis that required steroids or current pneumonitis. * Active infection requiring systemic therapy. * History or current evidence of any condition, therapy, or laboratory abnormality that might confound the results of the trial, interfere with the subject's participation for the full duration of the trial, or is not in the best interest of the subject to participate, in the opinion of the treating investigator. * Has known psychiatric or substance abuse disorders that would interfere with cooperation with the requirements of the trial. * Is pregnant or breastfeeding * Expecting to conceive or father children within the projected duration of the trial, starting with the pre-screening or screening visit through 120 days after the last dose of trial treatment. * Has received prior therapy with an anti-PD-1, anti-PD-L1, or anti-PD-L2 agent. * Has a known history of Human Immunodeficiency Virus (HIV) (HIV 1-2 antibodies). * Has known active Hepatitis B (eg, HBsAg reactive) or Hepatitis C (eg, HCV RNA \[qualitative\] is detected). * Has received a live vaccine within 30 days of planned start of study therapy. Note: Seasonal influenza vaccines for injection are generally inactivated flu vaccines and are allowed; however intranasal influenza vaccines (eg, Flu-Mist) are live attenuated vaccines, and are not allowed.