A Study to Learn About the Safety of Vedolizumab and How Well it Works in Children and Teenagers With Active Chronic Pouchitis

Inclusion Criteria: 1. The participant weighs \>=10 kg at the time of screening and first dose. 2. Has active chronic pouchitis, defined by a mPDAI score \>=5 assessed using the 3-day average of participant-reported clinical symptoms prior to the screening endoscopy (that is \[ie\] video pouchoscopy with biopsy) or bowel preparation for the endoscopy and a minimum mPDAI endoscopic subscore of 2 (outside the staple or suture line) and either: * \>=1 previous episodes of pouchitis within 1 year before the screening visit, with symptoms lasting for at least a total of 4 weeks, treated with \>=2 weeks of antibiotic or other prescription therapy (ie, other antibiotics, probiotics, immunomodulators, or anti-tumor necrosis factor \[TNFs\] within 1 year before screening). Or * Have had an inadequate response with, or lost response to, or be intolerant to antibiotic therapy (ie, requiring maintenance antibiotic therapy taken for \>=4 weeks immediately before the baseline endoscopy visit or not able to receive or continue antibiotic treatment due to intolerance or other contraindication). 3. The participant is aged 2 to 17 years, inclusive, at the time of screening and first dose. 4. The participant has a history of proctocolectomy and ileal pouch-anal anastomosis (IPAA) as treatment for ulcerative colitis (UC), Crohn's disease (CD), familial adenomatous polyposis (FAP), or other underlying conditions, such as Hirschsprung's disease, for which construction of a pouch was medically indicated, completed at least 1 year before the screening visit. Exclusion Criteria: The exclusion criteria are divided into 3 categories: active chronic pouchitis exclusion criteria, infectious disease exclusion criteria, and general exclusion criteria. Active Pouchitis Exclusion Criteria: 1. Has symptoms believed to be predominantly due to irritable pouch syndrome. 2. Has isolated cuffitis. 3. Is found to have dysplasia at the screening endoscopy. 4. Has mechanical complications of the pouch (for example \[e.g.\] pouch stricture or pouch fistula). 5. Currently requires or has a planned surgical intervention during the study. 6. Has a diverting stoma. Infectious Disease Exclusion Criteria: 7. Has evidence of an active infection (e.g. sepsis, cytomegalovirus \[CMV\], or listeriosis) during screening. 8. Had a clinically significant infection (e.g. pneumonia, pyelonephritis, coronavirus disease 2019 \[COVID-19\]) within 35 days before first dose of study drug. 9. Has active or latent tuberculosis (TB), as evidenced by a diagnostic TB test performed within 3 months of screening or during the screening period that is positive, as defined by: * A positive QuantiFERON test or 2 successive indeterminate QuantiFERON tests, or * A TB skin test reaction \>=5 millimeter (mm). NOTE: If participant have received Bacillus Calmette-Guérin vaccine, then a QuantiFERON TB Gold test should be performed instead of the TB skin test. NOTE: Participants with documented previously treated TB with a negative QuantiFERON test can be included in the study. 10. Has evidence of positive hepatitis B surface antigen (HBsAg) or hepatitis B core antibody (HBcAb). Hepatitis B virus (HBV) immune participants (e.g. HBsAg negative and hepatitis B antibody positive) may, however, be included. NOTE: If a participant tests negative for HBsAg, but positive for HBcAb, the participant would be considered eligible if the absence of HBV DNA is confirmed by HBV DNA polymerase chain reaction reflex testing performed in the central laboratory. 11. Has chronic hepatitis C virus (HCV) (ie, positive HCV antibody \[HCVAb\] and HCV Ribonucleic Acid \[RNA\]). NOTE: Participants who are HCVAb-positive without evidence of HCV RNA may be considered eligible (spontaneous viral clearance or previously treated and cured \[defined as no evidence of HCV RNA at least 12 weeks before baseline\]). 12. Has any identified congenital or acquired immunodeficiency (e.g. common variable immunodeficiency, HIV infection, organ transplantation). 13. Has positive stool studies for ova and/or parasites or stool culture at screening visit. 14. Has positive Clostridium difficile stool test at screening visit. General Exclusion Criteria: 15. Is taking, has taken, or is required to take any excluded medications. 16. Has active cerebral/meningeal disease, signs/symptoms, or history of progressive multifocal leukoencephalopathy (PML) or any other major neurological disorders, including stroke, multiple sclerosis, brain tumor, or neurodegenerative disease. 17. Has evidence of dysplasia or history of malignancy other than a successfully treated nonmetastatic cutaneous squamous cell or basal cell carcinoma or localized carcinoma in situ of the cervix. 18. Has any unstable or uncontrolled cardiovascular, pulmonary, hepatic, renal, GI, genitourinary, hematologic, coagulation, immunological, endocrine/metabolic, neurologic, or other medical disorder that, in the opinion of the investigator, would confound the study results or compromise participant safety.