You can check the balance without a 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.