If you do not have a picture ID, you may still recieve a replacement social security card if you have a current health insurance card with your name printed on it. This will suffice in proving your identity, but you will also need a document, like your US birth certificate to prove your citizenship. More information about obtaining a social security card can be found at http://www.ssa.gov/ssnumber/ss5doc.htm.
To get a social security card without a photo id, you will need a legal birth certificate with a seal or some other record that is legal proof of your citizenship. Possible substitutes are school records or military id cards.