Working in a private store, is it normal for the boss not to buy social security for employees?

This is not normal. Small shops are illegal.

According to the social insurance law, the unit must pay social insurance for employees, otherwise, the unit is illegal.

If the unit does not pay social insurance, employees can apply for labor arbitration to the district-level labor arbitration institute (located in the Human Resources and Social Security Bureau) where the unit is located and ask for social insurance. The union won the lawsuit.