Do I have to enroll an Apple Developer Program when working for a client?
You wouldn't have to buy a separate license, if you are a part of developer team on provisioning portal. Since you have a admin role, the certificates, devices and identifiers section should be visible to you. If you are not able to see them, probably the team agent has not yet accepted the latest terms and conditions released by Apple, with the new iOS and OSX.

You just need to contact the Agent of your team to sort this issue.