To change your credit card you must be logged in as the Account Owner, as only she or he has access to that section of the shop admin.

Once logged in as the Account Owner, go to the Account area and use the "Change or update" link beside the Payment Method.

A new page will appear prompting you for new card information. Once you've input this new info, click Confirm to continue and update your account. Shopify will automatically reset your billing attempts and process the card shortly thereafter.