I need to update my payment method. But it won't let me delete/remove the old payment card.

See -> Remove a payment method from your Apple ID

Remove a payment method from your Apple ID - Apple Support

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If you can't remove a payment method

If you have subscriptions like iCloud+, have an unpaid balance, or you use purchase sharing, you must keep at least one payment method on file.

• Do you have a subscription? You can cancel iCloud+ and cancel other subscriptions from Apple. When your subscription period is over, remove your payment method.
• If you're in a Family Sharing group and you use purchase sharing,
• the family organizer must have a payment method on file. If you're the
• family organizer, turn off purchase sharing, then remove the payment method.
• If you have an unpaid balance, you might not be able to remove a payment method. Instead, change your payment method.
• If you still can't remove a payment method, contact Apple Support.

Try this -> Sign in to your Apple ID account page and edit / update your Apple ID details.

https://appleid.apple.com                                                     

• On the Sidebar click Payment methods -> then tap Payments Methods and tap Manage payment method.

Or -> Add a payment method to your Apple ID

Add a payment method to your Apple ID – Apple Support (UK)

• If you want to change your card, you need to add first the new card, then remove the old card.
• or if you use your Apple ID balance, you need to set your Apple ID balance as your default payment.

If you update your payment method or add funds to your Apple ID balance, It would take about 1 to 3 business days in order to process.

If you're still unable to update your payment method, Contact Apple Support.

you need to add in a proper payment method and set that to your default form of payment. Some things won't work as the main form of payment like gift cards or prepaid cards, so you'll need to add in something new that they'll accept as a primary payment method first and then, once its accepted, set that to your default payment and make sure you've been charged by them on the new card before deleting the old card. sometimes I'll just add money to my Apple ID account with the new card info and try deleting again after that