A Study of Vedolizumab in Children With Ulcerative Colitis (UC) or Crohn's Disease (CD)

This multi-center trial is conducted worldwide. Up to 240 patients would be enrolled from Studies MLN0002-3024 \[participants with UC\] and MLN0002-3025 \[participants with CD\], either in the Treatment Cohort or in the Observational Cohort. Approximately 93 participants who have previously participated either in study MLN0002-3024 or MLN0002-3025, referred to as parent study, are expected to roll over to the MLN0002-3029 study in the treatment cohort. Treatment Cohort: The drug being tested in this study is called vedolizumab, being studied to treat pediatric patients who have UC or CD. Participants eligible for the Treatment Cohort can be administered vedolizumab intravenous (IV) at Week 54 visit of parent study or up to 1 week after Week 54 of the parent study based on the availability of test results needed to assess eligibility of the participant. At this study entry, participants will be administered the same blinded dose of vedolizumab IV that was received at Week 46 in the parent study and will then continue to receive vedolizumab IV at a frequency of once every 8 weeks (Q8W) in the following treatment groups: * Participants 10 to ≤15 kilogram (kg), Vedolizumab 150 milligram (mg) (High dose) * Participants 10 to ≤15 kg, Vedolizumab 100 mg (Low dose) * Participants \>15 to \<30 kg, Vedolizumab 200 mg (High dose) * Participants \>15 to \<30 kg, Vedolizumab 100 mg (Low dose) * Participants ≥30 kg, Vedolizumab 300 mg (High dose) * Participants ≥30 kg, Vedolizumab 150 mg (Low dose) Blinding of dose group assignment of the parent study will continue until the respective parent study is unblinded in order to protect the blinding of the parent study. The overall time to participate in the Treatment Cohort of this study is up to participant withdrawal, or until vedolizumab IV is commercially available for pediatric indication(s) in the participant's country or until other drug access programs become available, or Sponsor's decision for study closure, or for up to approximately 5 years, whichever comes first. Participants who complete or are discontinued from the study for any reason will complete the final safety/end of study (EOS) visit 18 weeks after their last dose of study drug. Observational Cohort: Participants who received at least 1 dose of study drug during parent study and early terminated or are not eligible for the Treatment Cohort of this study after completion of the Week 54 visit of parent study, will be enrolled in the Observational Cohort of this study as part of a long-term follow-up period to assess prespecified safety events of interest and will not receive continued treatment with vedolizumab IV. The overall time to participate in the Observational Cohort is up to approximately 2 years.