Phase-2 Study of Vedolizumab Plus Post-Transplant Cyclophosphamide and Short Course Tacrolimus for Graft-versus-Host Disease Prevention After Reduced Intensity Conditioning Peripheral Blood Stem Cell Allogeneic Hematopoietic Cell Transplantation

Inclusion Criteria: * Documented informed consent of the participant and/or legally authorized representative * Assent, when appropriate, will be obtained per institutional guidelines * Agreement to allow the use of archival tissue from diagnostic tumor biopsies * If unavailable, exceptions may be granted with study principal investigator (PI) approval * Age: ≥ 18 and ≤ 80 years old * Note: Patients \> 70 years of age must have Karnofsky performance status ≥ 80 and hematopoietic cell transplantation-comorbidity index (HCT-CI) ≤ 2 * Karnofsky performance status ≥ 70% * Patients with the following diagnosis, eligible to undergo allogeneic HCT from an 8/8 match related/unrelated donor (A, B, C, DR by high resolution typing) * Acute Leukemias (acute myeloid leukemia \[AML\] or acute lymphoblastic leukemia \[ALL\]) in complete remission with bone marrow (BM) blast of \< 5% * Myelodysplastic syndrome (blast \< 10%) * Myeloproliferative neoplasm (MPN) other than myelofibrosis (MF) needing HCT * Chronic myelomonocytic leukemia (CMML) * Hemoglobin ≥ 9g/dL (within 30 days prior to day 1 of protocol therapy) * NOTE: Red blood cell transfusions are not permitted within 14 days of hemoglobin assessment unless cytopenia is secondary to disease involvement * Total bilirubin ≤ 2.0 mg/dL (unless has Gilbert's disease) AND serum glutamic oxaloacetic transaminase (SGOT) and serum glutamic pyruvic transaminase (SGPT) \< 5 times the upper limit of normal (ULN) (within 30 days prior to day 1 of protocol therapy) * Aspartate aminotransferase (AST) =\< 3.0 x ULN (within 30 days prior to day 1 of protocol therapy) * Alanine aminotransferase (ALT) =\< 3.0 x ULN (within 30 days prior to day 1 of protocol therapy) * Creatinine clearance of ≤ 1.5 mg/dL or ≥ 60 mL/min per 24 hour urine test or the Cockcroft-Gault formula (within 30 days prior to day 1 of protocol therapy) * Left ventricular ejection fraction (LVEF) ≥ 50% * Note: To be performed within 28 days prior to day 1 of protocol therapy * IF ABLE TO PERFORM PULMONARY FUNCTION TESTS: Forced expiratory volume in 1 second (FEV1), forced vital capacity (FVC) and DLCO (diffusion capacity) ≥ 50% of predicted (corrected for hemoglobin) * Note To be performed within 28 days prior to day 1 of protocol therapy * IF UNABLE TO PERFORM PULMONARY FUNCTION TESTS: Oxygen (O2) saturation \> 92% on room air * Note To be performed within 28 days prior to day 1 of protocol therapy * Seronegative for HIV antigen/antibody (Ag/Ab) combo, hepatitis C virus (HCV), active hepatitis B virus (HBV) (surface antigen negative) (within 30 days prior to day 1 of protocol therapy) * HIV-infected patients on effective anti-retroviral therapy with undetectable viral load within 6 months are eligible for this trial * Tuberculosis test (within 30 days prior to day 1 of protocol therapy) * Patients with positive tuberculosis (TB) test results will have infectious disease (ID) evaluation and post HCT therapy with isoniazid (INH) for 6 months with ID follow up. Vaccinated patients will need negative chest X-ray results * Meets other institutional and federal requirements for infectious disease titer requirements * Note Infectious disease testing to be performed within 28 days prior to day 1 of protocol therapy * Women of childbearing potential (WOCBP): Negative urine or serum pregnancy test (within 30 days prior to day 1 of protocol therapy) * If the urine test is positive or cannot be confirmed as negative, a serum pregnancy test will be required * Agreement by females and males of childbearing potential to use an effective method of birth control or abstain from heterosexual activity for the course of the study through at least 3 months after the last dose of protocol therapy * 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: * Prior allogeneic HCT * Chemotherapy, radiation therapy, biological therapy, immunotherapy within 14 days prior to day 1 of protocol therapy * Note: Conditioning regimen within 14 days prior to day 1 of protocol therapy is not considered as an exclusion criterion. Patients on maintenance chemotherapy with agents listed are not excluded * Other investigational drugs for GVHD prophylaxis * Herbal medications * History of allergic reactions attributed to compounds of similar chemical or biologic composition to study agent * Clinically significant uncontrolled illness * Active infection not responding to antibiotics * Other active malignancy. Patients with a prior or concurrent malignancy whose natural history or treatment does not have the potential to interfere with the safety or efficacy assessment of the investigational regimen are eligible for this trial * Females only: Pregnant or breastfeeding * Patients not expected to be available for follow-up in our institution for at least 100 days after the transplant * 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 * 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)