In researching the matter further I find that I was mistaken. I was thinking of the Privacy Act of 1974, which stated: "no Federal, State, or local government could withhold a right, privilege or benefit from a person simply because the person refused to furnish his or her SSN."[1] Somehow I've believed for many years that it applied to private businesses as well, my mistake.

[1] https://www.ssa.gov/legislation/testimony_031606.html