Do I need a residence permit for my social security card?

No need. You can't apply for a social security card for a residence permit. You must have an ID card to apply for a social security card.

Those who meet the application conditions can make an appointment by telephone at the application outlets or apply for social security cards (including student ID cards) directly at the street (town) social security card application service outlets. When applying, you need to bring your ID card, household registration book, application form (citizens with collective household registration need to bring the household registration certificate issued by the local police station or police station) and other related materials.