Do I have to have an employment permit to apply for social security?

The employment certificate you mentioned should be the current national unified employment and unemployment registration certificate. The company requires employment certificate because with this certificate, the company can apply for tax exemption of 8,000 yuan per person per year, which is the national policy to promote employment. At present, many enterprises require employment certificates when recruiting employees for this policy, even as a threat of not handling social security and employment. This practice is suspected of defrauding state subsidies. You can complain to the labor department. As for training, it is the time to apply for employment unemployment registration certificate, and applicants required by the Social Security Bureau need to participate in unified vocational skills training.

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