Personalized Neoantigen Peptide-Based Vaccine in Combination With Pembrolizumab for Treatment of Advanced Solid Tumors

Inclusion Criteria COHORT 1 and COHORT 2 are no longer enrolling. PHASE I PRE-REGISTRATION, ALL: * Willing to provide tissue specimens per protocol, or have clinically collected formalin-fixed paraffin-embedded (FFPE) tissue blocks, or available sequencing data available from commercial or research tests * NOTE 1: Includes fresh tissue specimen at pre-registration, or with or clinically collected FFPE tissue blocks for complete exome and transcriptome sequencing. Patients who had tumor sequencing under certain Mayo IRB protocols and neoantigen has been identified or REAL Neo vaccine produced are allowed to proceed to pre-registration and/or registration. * NOTE 2: This criteria will not apply to patients who had sequencing and neoantigen prediction previously completed. * Measurable disease as defined by RECIST (v 1.1) criteria or non-measurable disease * NOTE: Tumor lesions in previously irradiated area are not considered measurable disease * Patients with actionable genomic abnormality including, but not limited to EGFR, ALK, MET, ROS-1, RET, NTRK, KRAS or BRAF must have received and progressed on at least one line of prior FDA-approved targeted therapy * Provide written informed consent. For pediatric patients (age 16-17 years): * Written informed consent from legal guardian(s) and/or child obtained in accordance with local regulations. * Willing to return to enrolling institution for follow-up * Willing to provide blood specimens for research * Negative pregnancy test 7 days prior to pre-registration for persons of childbearing potential. * Willing to employ highly effective method of contraception from pre-registration through 6 months after final vaccine cycle * Willing to receive tetanus vaccination if subject has not had one ≤ 1 year prior to pre-registration * Eastern Cooperative Oncology Group (ECOG) Performance Status (PS) of 0 or 1 * Anticipated life expectancy \> 6 months * Recovered from all toxicities associated with prior treatment to acceptable baseline status (see specified inclusion limits for laboratory toxicity) or National Cancer Institute (NCI) Common Terminology Criteria for Adverse Events (CTCAE) v5 Grade 0 or 1, except for toxicities not considered safety risk per treating investigator (e.g., alopecia or vitiligo). * The following lab values obtained ≤ 28 days prior to pre-registration: * Hemoglobin ≥ 9.0 g/dL (Must be ≥ 7 days after most recent transfusion) * Absolute neutrophil count (ANC) ≥ 1500/mm\^3 or ≥ 1.5 X 10\^9/L * Platelet count ≥ 100,000/mm\^3 or ≥ 100 X 10\^9/L (Must be ≥7 days after most recent transfusion) * Total bilirubin ≤ 1.5 x upper limit of normal (ULN) * Aspartate transaminase (AST) and alanine transaminase (ALT) ≤ 3 x ULN or ≤ 5 x ULN with liver metastases * Creatinine ≤ 1.5 x ULN OR calculated creatinine clearance must be ≥ 50 ml/min using Cockcroft-Gault formula * International normalized ratio (INR) or prothrombin time (PT) and activated partial thromboplastin time (aPTT) ≤ 1.5 x ULN unless patient is receiving anticoagulant therapy in which case PT or PTT must be within target range of therapy PHASE I REGISTRATION, ALL: * Successful sequencing and production of REAL-Neo vaccine * Measurable disease as defined by RECIST (v 1.1) criteria or non-measurable disease * NOTE: Tumor lesions in previously irradiated area are not considered measurable disease * ECOG PS 0 or 1 * Anticipated life expectancy \> 6 months * The following lab values obtained ≤ 14 days prior to registration: * Hemoglobin ≥ 9.0 g/dl * ANC ≥ 1500/mm\^3 * Platelet count ≥ 100,000/mm\^3 * Total bilirubin ≤ 1.5 x ULN * ALT and AST ≤ 3 x ULN (≤ 5 x ULN with liver involvement) * PT/INR and aPTT ≤ 1.5 x ULN unless patient is receiving anticoagulant therapy in which case INR or aPTT must be within target range of therapy * Calculated creatinine clearance ≥ 50 ml/min using Cockcroft-Gault formula * Provide written informed consent * Willing to provide blood and tissue specimens for research * Willing to return to enrolling institution for follow-up * Patients with actionable genomic abnormality including, but not limited to EGFR, ALK, MET, ROS-1, RET, NTRK, KRAS or BRAF must have also received and progressed on at least one line of prior FDA-approved targeted therapy * Negative pregnancy test ≤ 14 days prior to registration for persons of childbearing potential only * NOTE: If urine test cannot be confirmed negative, serum pregnancy test will be required * Willing to employ highly effective method of contraception from pre-registration through 6 months after final vaccine cycle * Willing to receive tetanus vaccination if subject has not had one ≤ 1 year prior to pre-registration * Recovered from all toxicities associated with prior treatment to acceptable baseline status or NCI CTCAE v5 Grade of 0 or 1, except for toxicities not considered safety risk per treating investigator (e.g., alopecia or vitiligo) PHASE II PRE-SCREENING COHORT 3 ONLY: * ECOG PS 0 or 1 * Histological confirmation of adenocarcinoma of the breast with estrogen receptor (ER) \< 10%, progesterone receptor (PR) \< 10%, and HER2 negative based on current American Society of Clinical Oncology (ASCO)/College of American Pathologists (CAP) guideline * Stage I-III based on 7th edition of TNM staging system from American Joint Committee on Cancer (AJCC) * Evidence of residual disease ≥ 1 cm after neoadjuvant pembrolizumab-based chemotherapy on imaging for patients who have not had surgery * Willing to proceed with surgery and provide tissue and blood specimens for patients who have not had surgery, willing to provide clinically collected FFPE tissue blocks, or willing to provide available sequencing data available from commercial or research test * Provide written informed consent * Willing to return to enrolling institution for follow-up PHASE II PRE-SCREENING COHORT 4 ONLY: * ECOG PS 0 or 1 * Histological confirmation of lung NSCLC * No actionable EGFR mutatio