Yes you can, but an institution may only issue you a card as long as the collateral (the cash to secure it) matches the amount that is desired for the credit card.

In the US, banks and the institutions that are behind department store credit cards require Social Security numbers. As the Social Security is the only way to track your identity for sure, these financial bodies need your social security number to reference previous credit history.

There are several companies that offer a Pre-Paid Debit Card, which is not exactly a credit card but at least you can get a card with a credit card logo on it.  Since the Patriot Act, no banks will issue a bank account, or a credit card without having a Personal Identification Number.