Recurly provides support for clearing your customer data from the system if requested to stay within the GDPR guidelines. If you are an admin user, you can easily do this by going to the customer's account page in Recurly. From there, click on "Edit" near their billing information, and once you've been redirected, you should be on a page where you can change their billing info. In the top right-hand corner of the page should be a red "Clear Information" button. Clicking on that button will remove the customer's credit card and related billing address information from the account.
Otherwise, if you are not an admin, you will need to close the customer's account which will terminate their subscription and also clear the billing information as well. To do this, you'll want to access their account page, then click on "Account Actions" in the top right-hand corner of the page and click on "Close Account". Doing this will close the account, deleting any currently stored billing information and canceling active subscriptions. The close account status will stop any new purchases from being made on the account unless reopened in the future.