Social security SMS query balance without social security card

Yes, but you should provide your own social security number.

Your social security number, the company staff can check it for you.

If you haven't applied for a social security card before, you can take your ID card to a bank that is qualified to apply for a social security card and open it yourself. Conditional banks can ask the staff of your company or the local social security hall. Very convenient.

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