Inside the Billing menu, there is no point where you can manage your card details because we do not store them.
In case you are in the free plan:
To renew your subscription, you'll need to manually select the plan and billing cycle you wish to purchase, and proceed with the payment using a new credit/debit card (just like you did the first time you bought a subscription). At the next renewal, the system will charge the new card without any further action on your behalf.
If your subscription has not expired yet:
Before proceeding with the purchase, you need to make sure that your subscription is not active (it has an expiration date). Otherwise, you should click on the "Cancel subscription" option on the subscription page. This will stop the program from trying to charge you automatically.
Subscription cancellation does not mean you will lose your data or your subscription will expire (or your organization will be downgraded) before a predetermined date expiration.
The system makes 3 attempts to charge you before downgrading your account to the free plan. To select your new current plan again, you will need to cancel your subscription, otherwise it does not appear as an option to buy.
Then, you can choose the plan you are interested in, the billing cycle (annual / monthly) and click Review order. You will then be asked to submit your credit card details and once the purchase is complete, the system will keep these card details for the next auto-renewal.