Multitargeted Recombinant Ad5 PSA/MUC-1/Brachyury-Based Immunotherapy (TriAdeno) Vaccine With IL-15 Superagonist N-803 in Participants With Clinically Localized Prostate Cancer Undergoing Active Surveillance

* INCLUSION CRITERIA: * Histologically confirmed diagnosis of organ confined, low- or intermediate-risk PCa (Gleason grade group 1 or 2) identified in at least one prostate biopsy core. Biopsies performed at outside institutions should have Gleason score confirmed at the NCI by a genitourinary (GU) pathologist. * Participants must be on active surveillance. * Pre-study treatment tissue availability (at least one formalin-fixed paraffin embedded \[FFPE\] biopsy core or one H\&E-stained slide and at least 5 unstained slides) obtained between 3 and 24 months prior to treatment initiation is mandatory for study initiation. * Serum PSA level of \<20 ng/mL (or \<10ng/mL for participants being treated with 5- alpha-reductase inhibitors) * Clinical stage \<=T2a by digital rectal exam (DRE) * Age \>=18 years * Eastern Cooperative Oncology Group (ECOG) performance status \<=1. * Adequate organ and marrow function as defined below: * Absolute neutrophil count (ANC) \>=1.0 x 10\^9/L * Hemoglobin (Hgb) \>=9 g/dL * Platelets \>=75,000/mcL * Prothrombin International Normalized Ratio (INR) \<1.5 x upper limit of normal (ULN) * Partial thromboplastin time (PTT) \<1.5 x ULN * Total bilirubin \<1.5 x ULN * Aspartate aminotransferase (AST) \<=2.5 x ULN * Alanine aminotransferase (ALT) \<=2.5 x ULN * Creatinine \<=1.5 x ULN OR --Calculated Creatinine clearance \>=40 mL/min/1.73 m\^2 for individuals with creatinine levels above institutional normal (using either Chronic Kidney Disease Epidemiology Collaboration \[CKD-EPI\] equation) * Treatment with steroid therapy must have a washout of at least 6 weeks prior to initiation of study treatment. Physiologic (replacement) doses of steroids as well as nasal, topical, or inhaled steroids are allowed. * Vaccination with a live (attenuated) vaccine (e.g., FluMist(R)) or a killed (inactivated)/subunit vaccine (e.g., PNEUMOVAX(R), Fluzone(R)) must occur not sooner than 28 days or 14 days, respectively, prior to initiation of study treatment. * Participants must agree to use an effective method of contraception (barrier, surgical sterilization, abstinence) for the duration of the study treatment and up to one (1) month after the last study drug injection. Participants must not freeze or donate sperm within the same period. * Participants must be able to understand and willing to sign a written informed consent document. EXCLUSION CRITERIA: * Prior treatment for PCa by surgery, radiation, local ablative (i.e., cryosurgery or highintensity focused ultrasound), or androgen-deprivation therapy. * Evidence of PCa with metastatic disease. * Prior treatment with adenovirus-based vector immunotherapy, adenovirus-based vaccines, or investigational vaccines. * Prior solid organ or bone marrow transplant. * Immunodeficiency or splenectomy. * Presence of a known active acute or chronic infection, including human immunodeficiency virus (HIV), confirmed by PCR, and hepatitis B virus (HBV) and hepatitis C virus (HCV), as determined by hepatitis B surface antigen (HBsAg) and HCV serology. * History of autoimmune disease (active or past), except for autoimmune-related thyroid disease, type I diabetes, and vitiligo if the condition(s) is well controlled. * History of heart disease, such as congestive heart failure (class II, III, or IV defined by the New York Heart Association functional classification), history of unstable or poorly controlled angina, or history (\<1 year prior to initiation of study therapy) of ventricular arrhythmia. * Acute or chronic skin disorders that will interfere with injection into the skin of the extremities or subsequent assessment of potential skin reactions. * Second malignancy within 3 years prior to initiation of study therapy. Note: Individuals with curatively treated non-melanoma skin cancers or non-muscle invasive bladder cancer will not be excluded. * History of herbal products that may decrease PSA levels (e.g., saw palmetto). * Participants who have undergone surgery within 4 weeks prior to initiation of study therapy. * Participants receiving any other investigational agents within 30 days prior to initiation of study therapy. * History of allergic reaction attributed to compounds of similar chemical or biological composition to the study drugs. * Uncontrolled intercurrent illness that would limit compliance with study requirements suggested by medical history, physical examination, or standard clinical assessments such as imaging, EKG, and laboratory studies.