Yes.  You will always have to give your personal information, including your social security number, when applying for any type of credit card.  However with a prepaid card, you will not receive an itemized bill, unless you request one, and the account associated with the card will not show up on your credit report.  On the merchant side, your purchases will still be traceable. 

You can build up your available balance on your prepaid card simply by making cash deposits at most major grocery stores and gas stations.  A prepaid card works a lot like a debit card - once the money is gone, it’s gone.  A prepaid card will not allow you to charge over the available balance and there are no bills or interest fees.