I realize this question (or comment) is a year old, but given that many people have viewed the question in the last year, I am going to answer it.
The question is based on an erroneous interpretation of the Privacy Act. The Privacy Act states, in relevant part:
“It shall be unlawful for any Federal, State or local government agency to deny to any individual any right, benefit, or privilege provided by law because of such individual's refusal to disclose his social security account number.”
The Privacy Act also has some exceptions that are inapplicable here. However, what should be clear is that the Privacy Act states the exact opposite of what is claimed in the question. Since the credit bureaus are NOT government agencies, the privacy act does not apply to them. Accordingly, the credit bureaus may ask for your social security number.
Moreover, it should be pointed out that the credit bureaus are not the entities asking for your social security number. When you apply for credit, the creditor is asking for your social security number. Since creditors are usually private businesses, and not government agencies, the Privacy Act simply does not apply to them.