The company address has moved. Do employees need to go to their previous social security bureau?

The company relocation has nothing to do with social security. Social security needs to be paid normally, or by the social security bureau where the company is registered.

According to the relevant provisions of the Social Insurance Law, the employer shall, within 30 days from the date of its establishment, apply to the local social insurance agency for social insurance registration with its business license, registration certificate or unit seal. The social insurance agency shall, within fifteen days from the date of receiving the application, examine and issue the social insurance registration certificate.

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