We provide IT services to a group of customers. We have a Data Center. Can just our ITSM Group get Certified, or does the entire organization need to be Certified? Would a Data center have different requirements?
You can certify only one part of your organization, like you mentioned ITSM group. That will be your scope.
This article will help you regarding scope definition: "How to define the scope of the SMS in ISO 20000" https://advisera.com/20000academy/blog/2015/06/02/how-to-define-the-scope-of-the-sms-in-iso-20000/
Data Center has to fulfill all requirements of the ISO 20000 (if that would be your scope). ISO 20000 does not set any particular requirements towards any part of the organization.