Your Question
Is it legal for a company to do a credit check without having a full social security number?
Of course, unless they violate your privacy rights.
What would be illegal about it? How do you suppose credit checks were done prior to the creation of Social Security in about 1933?
It is illegal to do a credit check without your written permission. You can be identified with less personal data than you think. Probably just your IP address (the internet phone number of the computer you are using) could be used to derive as much information about you as any credit check.
In addition to the other attorney answers, it's not really possible to do any kind of in-depth credit check without a social security number and date of birth. Consult with an attorney in your area for specific advice.
Kevin B. Murphy, B.S., M.B.A., J.D. - Mr. Franchise
Franchise Attorney
It's only legal to run a credit check under limited circumstances. Those circumstances have nothing to do with whether the company has the individual's complete social security number. Assuming that the company can legally check the person's credit, having an incomplete SSN would only cause practical problems, not legal ones.