The short answer is no. The right to work legally in the United States is one of the rights that comes with permanent residence.
As a permanent resident (green card holder), you have the right to live permanently in the USA and work in the USA at any legal work of your qualification and choosing (except for some jobs limited to US citizens only for security reasons).
US citizens and lawful permanent residents do not need an Employment Authorization Document (EAD) or any other working permit, other than their green card.
However, if your application for the green card is still pending you will have to file Form I-765, Application for Employment Authorization, in order to obtain the right to work in the USA till you receive your green card. You can file Form I-765 either together with or after filing of Form I-485, Application to Register Permanent Residence or Adjust Status.