It can be done. but it is not a good solution (from a security standpoint).
On the Windows platform
This assumes you have mapped your CAC/PIV cert to your privileged AD account.
Your workstation client running the web browser must be CAC/PIV enabled.
Run the web browser with your elevated credentials, using your CAC/PIV.
Open the connection to your vCenter and click "Use Windows Session Authentication. (Client integration plug-in required).
You should now be logged in to your vCenter using your CAC/PIV. (Be warned Don't browse the Internet with this web browser session, extreme danger!!!)
Maybe someone else knows a safer way.
Charles