Does it take half a year to get a social security card?

If it is a local, the street will issue social security cards in a unified manner.

If you are a foreigner, there is no social security card, only a medical insurance card. You can ask the staff of the unit to go to the medical insurance bureau in the district where the unit is located or bring your ID card to get it. Almost like you said, you can get a medical insurance card after adding money for almost half a year.