A Vaccine (CMV-MVA Triplex Vaccine) for the Enhancement of CMV-Specific Immunity and the Prevention of CMV Viremia in Patients Undergoing Haploidentical Hematopoietic Stem Cell Transplant

Inclusion Criteria: * DONORS: Documented informed consent of the participant. This can be done in person or informed consent can be obtained remotely. * Remote consent, when appropriate, will be obtained per institutional guidelines. * Assent, when appropriate, will be obtained per institutional guidelines. * Adult subjects who require a legally authorized representative (LAR) will not be permitted to be enrolled under this protocol. * DONORS: Age: 18 - 75. * DONORS: Women of childbearing potential (WOCBP): negative urine or serum pregnancy test. If the urine test is positive or cannot be confirmed as negative, a serum pregnancy test will be required. * DONORS: Agreement by females and males of childbearing potential\* to use an effective method of birth control (hormonal or barrier method) or abstain from heterosexual activity prior to study entry and for up to 90 days post-vaccination. * Childbearing potential defined as not being surgically sterilized (men and women) or have not been free from menses for \> 1 year (women only). * RECIPIENTS: Documented informed consent of the participant and/or legally authorized representative. This can be done in person or informed consent can be obtained remotely. * Remote consent, when appropriate, will be obtained per institutional guidelines. * Assent, when appropriate, will be obtained per institutional guidelines. * Adult subjects who require a legally authorized representative (LAR) will not be permitted to be enrolled under this protocol. * RECIPIENTS: Participant must be willing to comply with study and/or follow-up procedures, including willingness to be followed for one year post-HCT. * RECIPIENTS: Age: 18 - 75. * RECIPIENTS: Planned peripheral blood stem cell (PBSC) or bone marrow (BM) HCT for the treatment of the following hematologic malignancies: * Lymphoma (Hodgkin and Non-Hodgkin). * Myelodysplastic syndrome. * Acute lymphoblastic leukemia in first or second remission (for acute lymphoblastic leukemia/lymphoblastic lymphoma, the disease status must be in hematologic remission by bone marrow and peripheral blood. Persistent lymphadenopathy on computed tomography (CT) or CT/positron emission tomography(PET) scan without progression is allowed.) * Acute myeloid leukemia in first or second remission. * Chronic myelogenous leukemia in first chronic or accelerated phase, or in second chronic phase. * Other hematologic malignancies judged appropriate by the clinical principal investigators (PIs), including chronic lymphocytic leukemia, myeloproliferative disorders and myelofibrosis. Patients with multiple myeloma and those with non-malignant disease such as aplastic anemia are excluded\*\*. * Adult cases of multiple myeloma (MM) are excluded as HCT is not standard of care for MM and is only performed in very advanced cases with an associated high risk of relapse and non-relapse mortality (NRM). Adults with aplastic anemia are excluded because their standard management includes T cell depletion with agents such as antithymocyte globulin (ATG), which is not permissible on this protocol. Patients undergoing a second haploHCT are not eligible (patients who have undergone a previous autologous HCT are eligible). * RECIPIENTS: Patients receiving myeloablative (MA) or reduced intensity conditioning (RIC) are allowed. * RECIPIENTS: CMV seropositive. * RECIPIENTS: Eligible haploidentical donors will have 2-4 mismatches if human leukocyte antigen (HLA)-A, -B, -C, and -DRB1 typing is used; 2-5 mismatches if HLA-A, -B, -C, -DRB1, and -DQB1 typing is used; and 2-6 mismatches if HLA-A, -B, -C, -DRB1, -DQB1, and -DPB1 typing is used. A unidirectional mismatch in either the graft versus host or host versus graft direction is considered a mismatch. The donor and recipient must demonstrate that they are a full haplotype match by being identical at a minimum of one allele (at high resolution deoxyribonucleic acid \[DNA\]-based typing) at the following genetic loci: HLA-A, -B, -C, and DRB1 if 8 allele typing is used; HLA-A, -B, -C, -DRB1, and -DQB1 if 10 allele typing is used; and HLA-A, -B, -C, -DRB1-, DQB1, and -DPB1 is 12 allele typing is used. * RECIPIENTS: Planned HCT with minimal to no-T cell depletion of graft. * RECIPIENTS: Conditioning and immunosuppressive regimens according to institutional guidelines are permitted. * RECIPIENTS: Total bilirubin ≤ 1.5 X upper limit of normal (ULN) (unless has Gilbert's disease) (to be performed within 45 days prior to day 1 of protocol therapy). * RECIPIENTS: Aspartate aminotransferase (AST) =\< 2.5 x ULN (to be performed within 45 days prior to day 1 of protocol therapy). * RECIPIENTS: Alanine aminotransferase (ALT) =\< 2.5 x ULN (to be performed within 45 days prior to day 1 of protocol therapy). * RECIPIENTS: Estimated creatinine clearance acceptable per institutional guidelines (to be performed within 45 days prior to day 1 of protocol therapy). * RECIPIENTS: Left ventricular ejection fraction (LVEF) ≥ 50%. * Note: To be performed within 45 days prior to day 1 of protocol therapy. * RECIPIENTS: If able to perform pulmonary function tests: forced expiratory volume in 1 second (FEV1), forced vital capacity (FVC) and carbon monoxide diffusing capability (DLCO) (diffusion capacity) ≥ 50% of predicted (corrected for hemoglobin). * If unable to perform pulmonary function tests: Oxygen (O2) saturation \> 92% on room air. * Note to be performed within 45 days prior to day 1 of protocol therapy. * RECIPIENTS: Seronegative for HIV antigen (Ag)/antibody (Ab) combination (combo), hepatitis c virus (HCV)\*, active hepatitis b virus (HBV) (surface antigen negative) and syphilis (RPR) within 2 months of registration and no history of disseminated cutaneous human papillomavirus (HPV) related disease. * If seropositive for HIV, HCV or HBV, nucleic acid quantitation must be performed. Viral load must be undetectable. * RECIPIENTS: Meets other institutional and federal requirements for infectious disease titer requirements. * Note Infectious disease testing to be performed within 45 days prior to day 1 of protocol therapy. * RECIPIENTS: Women of childbearing potential (WOCBP): negative urine or serum pregnancy test. If the urine test is positive or cannot be confirmed as negative, a serum pregnancy test will be required (to be performed within 45 days prior to day 1 of protocol therapy). * RECIPIENTS: Agreement by females and males of childbearing potential\* to use an effective method of birth control (hormonal or barrier method) or abstain from heterosexual activity prior to study entry and up to 90 days post-HCT. * Childbearing potential defined as not being surgically sterilized (men and women) or have not been free from menses for \> 1 year (women only). Exclusion Criteria: * DONORS: Any prior transplant to day 1 of protocol therapy (day 1 defined as the day after donors receive the Triplex vaccine). * DONORS: Chemotherapy, radiation therapy, biological therapy, immunotherapy within 21 days prior to day 1 of protocol therapy. * DONORS: Receipt of any vaccine (licensed or investigational) within 30 days prior to and after the study vaccine. * DONORS: Unfit to undergo standard stem cell mobilization and apheresis e.g. abnormal blood counts, history of stroke, uncontrolled hypertension. * DONORS: Sickling hemoglobinopathy including hemoglobin (Hb)SS, HbAS, HbSC. * DONORS: Donors with impaired cardiac function are excluded. Electrocardiography is routine for potential HCT donors over 60 years old and those with a history of heart disease. Subjects in whom cardiac function is abnormal (excluding 1st degree branch block, sinus bradycardia, sinus tachycardia or non-specific T wave changes) are ineligible for Triplex vaccination. * DONORS: Severe psychiatric illness. Mental deficiency sufficiently severe as to make compliance with the donation procedure unlikely and making informed consent impossible. * DONORS: Females only: Pregnant or breastfeeding. * DONORS: Any other condition that would, in the investigator's judgment, contraindicate the patient's participation in the clinical study due to safety concerns with clinical study procedures. * DONORS: Prospective participants who, in the opinion of the investigator, may not be able to comply with all study procedures (including compliance issues related to feasibility/logistics). * RECIPIENTS: Any prior investigational CMV vaccine. * RECIPIENTS: Experimental anti-CMV chemotherapy in the last 6 months. * RECIPIENTS: Live attenuated vaccines (planned medications from the time of HCT to day 70 post-HCT). * RECIPIENTS: Medically indicated subunit (Engerix-B for HBV; Gardasil for HPV) or killed vaccines (e.g. influenza, pneumococcal, or allergy treatment with antigen injections) (planned medications from the time of HCT to day 70 post-HCT). * RECIPIENTS: Allergy treatment with antigen injections (planned medications from the time of HCT to day 70 post-HCT). * RECIPIENTS: Alemtuzumab or any equivalent in vivo T-cell depleting agent (or CD34+ selection) (planned medications from the time of HCT to day 70 post-HCT). * RECIPIENTS: Antiviral medications with known therapeutic effects on CMV such as ganciclovir (GCV)/valganciclovir (VAL), foscarnet (FOS), cidofovir, CMX-001, maribavir. Acyclovir has no known therapeutic efficacy against CMV and is allowable as standard of care to prevent herpes simplex virus (HSV) (planned medications from the time of HCT to day 70 post-HCT). * RECIPIENTS: Prophylactic therapy with CMV immunoglobulin or prophylactic antiviral CMV treatment EXCEPT letermovir prophylaxis (prior to day 100) (planned medications from the time of HCT to day 70 post-HCT). * RECIPIENTS: Disease-based radiation therapy (not total body irradiation) (planned medications from the time of HCT to day 70 post-HCT). * RECIPIENTS: Other investigational product(s) - concurrent enrollment in other clinical trials using any investigational new drug (IND) drugs with unknown effects on CMV or with unknown toxicity profiles is prohibited (planned medications from the time of HCT to day 70 post-HCT). * RECIPIENTS: Other medications that might interfere with the evaluation of the investigational product (planned medications from the time of HCT to day 70 post-HCT). * RECIPIENTS: Patients with active autoimmune conditions requiring systemic immunosuppressive therapy within the previous 5 years. * RECIPIENTS: Patients considered by PI/consenting physicians to have a complicated prior therapy or HCT regimen, or who have a low survival probability (e.g., refractory leukemia and/or undergoing 2nd HCT). * RECIPIENTS: Poor risk disease/disease status including: Chronic myelogenous leukemia (CML) in blast crisis, acute myeloid leukemia (AML)/acute lymphoblastic leukemia (ALL) beyond 2nd remission, multiple myeloma, and aplastic anemia. * RECIPIENTS: Females only: Pregnant or breastfeeding. * RECIPIENTS: Any other condition that would, in the investigator's judgment, contraindicate the patient's participation in the clinical study due to safety concerns with clinical study procedures. * RECIPIENTS: Prospective participants who, in the opinion of the investigator, may not be able to comply with all study procedures (including compliance issues related to feasibility/logistics).