Click Account (head icon) in the upper right menu and choose Purchases in the drop down menu.  There you can delete a credit card associated with an event.

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FOR FLIGHT ADMINISTRATOR:  You can assist a user in deleting a credit card on file.  Go to Contacts > Quick Checkin tab.  The far-right column is "CC Req".  If a credit card is on file you will see a trash can/delete icon.

Important:  If you do not see a CC Req column, it is because a credit card is not required for the flight.  Financials > Setup tab.