Here is Apple's instructions https://support.apple.com/en-us/118293 and worth noting is that they include a "If there's an issue, change your payment method". They seem to be suggesting that if you have a problem changing the method, try adding a method then remove the old method. I have encountered this issue with other online accounts, they seemingly wont let you remove or edit an existing payment method until you enter another one. I assume this is because you cannot have an Apple Account without a payment method.
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