I had to actually go the local office (could not be done online) to submit a form. You fill out the form to request a replacement social security card (form available on-line). I showed them my brand new US passport, old card, and state issued driver's license, the lady verified everything and it took about a week to get a new card in the mail. In my case, my old card still said "Valid for work only with INS authorization" so I definitely needed a new physical card.
I don't think there is a whole lot you can do with the social security system without visiting a local office - which causes the other issue with this...long wait times. They want most people to stop by to ask their questions but they are not fast.
I went on a Thursday morning, mid month and was there 20 minutes before they opened. I was #19 in line and ended up waiting over an hour. Once called up, the whole process took 3 minutes since I had my form filled out and my evidence ready.