No, you don’t need a work permit. Permanent residence grants a right to work legally in the USA by default.
Green card holders can live in the USA on a permanent basis and be employed at any legal job according to their choice and qualifications (with the exception of some fields available only for American citizens due to national security reasons).
Employment Authorization Document (EAD) or any other work permit is not required both from US citizens and lawful permanent residents (who need only their green card).
However, you have to submit Application for Employment Authorization (Form I-765) if your green card application is still processing. In this case you need to get the right to work legally until your green card is ready. Form I-765 may be submitted either together with or after filing of Application to Register Permanent Residence or Adjust Status (Form I-485).
February 3, 2019