Vaccine Therapy and Pembrolizumab in Treating Patients With Solid Tumors That Have Failed Prior Therapy

• Since p53 mutations occur in a wide variety of tumor types, this is a mixed histology study for incurable tumors; subjects with the following solid tumors are eligible for screening: non-small cell lung cancer, squamous cell carcinoma of the head and neck, hepatocellular carcinoma, renal cell carcinoma, melanoma, bladder, soft tissue sarcoma, triple-negative breast cancer, and colorectal carcinoma displaying microsatellite instability and pancreatic cancer
• Advanced (unresectable) solid tumors: patients must have failed or been intolerant to at least one line of standard therapy or refuse standard treatment
• Performance status: patients must have an Eastern Cooperative Oncology Group (ECOG) =\< 2 (Karnofsky >= 60%)
• Informed consent: all subjects must have the ability to understand and the willingness to sign an Institutional Review Board (IRB) approved consent form
• Absolute neutrophil count: >= 1,500/ul
• Platelets >= 100,000/ul
• Hemoglobin level: must be greater than 9 g/dL
• Renal function: calculated or measured creatinine clearance >= 50 ml/min and/or serum creatinine =\< 1.6 mg/dl
• Total bilirubin =\< 1.5 x institutional upper limit of normal
• Aspartate aminotransferase (AST) and alanine aminotransferase (ALT) =\< 3 times institutional upper normal level (AST and ALT =\< 5 times institutional upper normal level, if there is evidence of liver metastasis)
• Confirmed p53 involvement: patients with p53 over-expression by immunohistochemistry (>= 10% of cells within the tumor staining positive) or those with a p53 mutation as determined by mutational analysis of tumor tissue will be eligible; patients with prior exposure to p53-based vaccines will be eligible
• Agreement to use adequate contraception: women of child-bearing potential must use contraception prior to study entry and for six months after study participation; men that are sexually active whose partners are women of childbearing age must use condoms

Exclusion Criteria:

• Patients may not be receiving any additional investigational agents or radiation therapy
• Pregnancy: pregnant women are excluded from this study; should a woman become pregnant or suspect that she is pregnant while participating on the trial, she should inform her treating physician immediately; women who are pregnant or breastfeeding are excluded
• Patients with known brain metastasis
• Radiotherapy within 4 weeks prior to entering the study
• Patients with previous exposure to anti-programmed cell death (PD)-1 or anti-programmed cell death ligand 1 (PDL-1) will not be eligible
• History of allergy to egg proteins
• Patients who have not recovered from adverse events due to agents administered more than 4 weeks earlier
• Concurrent use of systemic corticosteroids (nasal corticosteroids, inhaled steroids, adrenal replacement steroids, and topical steroids are allowed)
• History of immunodeficiency or autoimmune disease: patients with a history of immunodeficiency, including organ grafts and human immunodeficiency virus (HIV), will not be eligible
• Patients with a history of autoimmune disease will also be excluded, specifically those with any active autoimmune disease or a condition that requires systemic corticosteroids; exceptions to this are subjects with vitiligo and type I diabetes mellitus, who will be permitted to enroll
• Patients with a history of severe immune-mediated adverse reactions with ipilimumab: this will be defined as any grade 4 toxicity requiring treatment with corticosteroids (greater than 10 mg/day prednisone or equivalent dose) for greater than 12 weeks
• Patients with a history of cardiac disease are excluded; baseline electrocardiography and assessment of serum troponin (I) are included the screening exams; subjects in whom these assays are abnormal (electrocardiogram [EKG] excluding 1st degree bundle branch block, sinus bradycardia, sinus tachycardia or non-specific T wave changes, serum troponin >= grade 2) are ineligible
• Non-compliance: if it is the opinion of the investigator that a subject may be unable to comply with the safety monitoring requirements of the study, they will be excluded