To validate you did save the new card, delete an old/expired card or to designate that newly vaulted credit card for the Auto-Renewal process, check your My Credit Card Vault location. The link to one's My Credit Card Vault will be found on the Dashboard of their CM account.
When logged into your CM account, click on the small arrow by your name in the upper-left of the screen.
Next, click "My Account" from the drop-down that'll appear to be brought to your Dashboard.
There, you'll see a link to My Credit Card Vault, which will bring you to the Vault when clicked.