Yes. I just got married and have taken my husband's last name.
To obtain a new card, you must first contact Social Security.
Why is contacting Social Security necessary to receive a new Social Security card?
If you do not contact Social Security to update your name change it could prevent your work wages from being correctly posted to your Social Security record. This could lower the amount of Social Security you receive in the future. It could also cause delays when you file your taxes.