ANSWER: Sure, as a matter of constitutional law. There is no constitutional constraint on what private persons do (other than the Thirteenth Amendment).
QUESTION: I suppose my concrete example would be not allowing out-of-state licenses to serve for purposes of alcoholic or drug related purchases. Is that permissible?
ANSWER: If this is simply the policy of a private actor, sure, no constitutional problem there.
QUESTION: What about not allowing passports to serve as identification?
ANSWER: Again, no constitutional problem. There may be statutory obligations imposed on private actors that make this sort of think illegal. But it would not be a constitutional problem.
QUESTION: It seems to me these run in the face of the clause, except that these are not state actors. Is there a difference?
ANSWER: Yes, a big one. Constitutional law constrains the government (only).