It seems like you have to enable the Google Play Family Library from the family settings.

Go to https://myaccount.google.com/family/details

And activate the Google Play Store Family Library
(might work with directly navigating to https://play.google.com/store/family/create)

In the process of activating the library you can select a payment method.

After that is done, a new link "Change family payment method" / "Familienzahlungsmethode ändern" appears under the Play Store Payment Methods:
https://play.google.com/store/paymentmethods


Other helpful resources:
* https://support.google.com/googleplay/thread/9850356/google-family-payment-method-not-available?hl=en
* https://support.google.com/families/answer/7007852?hl=en
* https://support.google.com/googleplay/answer/6294544?hl=en#zippy=%2Cadd-a-family-payment-method

Some search keywords:
- Add google play family payment methods from iPhone
- Add google play family payment moethod without Android device or Google Play app