Does social security have a social security card now?

Generally speaking, social security card is actually a medical insurance card. If you only buy old-age care and don't buy medical care, the Social Security Bureau won't give you a medical insurance card.

If you can still see a doctor, it usually takes one month for the Social Security Bureau to apply for a card. Under normal circumstances, you can get the card in one month. You can ask the personnel of the company. If you can't find it, you can only reissue a social security card and a copy of your ID card.

Further reading: How to buy insurance, which is good, and teach you how to avoid these "pits" of insurance.