Does “consumer co-op” mean that you have to buy a plastic card in order to get the corporation money, like Costco?

No, Costco is a public company that sells annual memberships and then retails to its members at prices that are just about break even not counting the membership fee. The shareholders of the company elect the board of directors who runs the company.

REI is owned by people who pay a one-time lifetime membership fee. Those co-op members elect the board fo directors who runs the company.