Customers can update their payment information by clicking "Change payment information" within the customer portal. This will send an email from Shopify to update the credit card information.
More details on how a customer can update their payment information can be found in our Help Center article: https://support.boldcommerce.com/hc/en-us/articles/360055620132-Subscription-Management-Customer-Perspective#update-payment-method-or-billing-address