How to use a table when the user identifier can be one of two fields
When doing this sort of design, I think first about the "things", and then about the "relationships".
The things are:


the people
the credit cards and
the organisation.


The relationships are:


Who owns the organisation.
Who is a member of the organisation.
Which credit card is the owner using to pay the bill


You need an entity for each "thing".
Depending on whether it is a "to one" relationship or a "to many" relationship, you can either define the relationship within an entity or you will need a new entity for the relatonship.

Who owns the organisation: If only one owner, you can put the owner in the organsation entity, otherwise you would need a separate entity.
Who is a member of the organisation: You can have many members of an organisation, therefore you will need a separate entity.
Which credit card is the owner using to pay the bill: Only one credit card would be used therefore you can put the reference to the credit card card in the organisation entity.

So you would have four entities:
Person (a "Thing")
Credit Card (a "Thing")
Organisation (a "Thing", with two relationships)
Organisation_Member (a relationship)

Here are some example columns you could use for the entities

Person Columns:


person_id (primary key, identity)
name
email


Credit Card Columns


person_id
billing_id (start from 1, don't make it an identity)
+credit card information columns (I expect you would encrypt the data, so not store the actual card numbers as text)


(The combination of person_id and billing_id in the above forms the primary key)

Organisation Columns:


org_id
org name
person_id (of the (one) owner - defining one relationship mentioned)
billing_id (of the (responsible) owner - defining another relationship mentioned)


Organisation_Member:


org_id
person_id (of the member)


One more point about email addresses. If you want there to be a separate email address for the organisation, it can be included in the organisation table. I'm not sure I understand why you would need this. Perhaps my answer would be different if I understood what it is for.