Do I have to enroll an Apple Developer Program when working for a client? The problem: I cannot see certificates section in client's member center.

Disclaimer: I am new to iOS platform so please bear it in mind.

My client has set up an Apple Developer account at https://developer.apple.com/membercenter/ . His account is paid and valid (he already published something). He has also an iTunes Connect account at https://itunesconnect.apple.com/.

I have only iTunes Connect account. (Didn't pay $100 for my own Developer Program).

Next, the client added me to his Apple Developer account and granted me a "team admin" role. But I still can see only 2 sections at https://developer.apple.com/membercenter: SDKs and Bug Reporting.

Do I have to buy a Developer Program for myself to generate certificates?
Or there can be other reasons for this problem?
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.