A work permit is formally known as an Employment Authorization Document (EAD). Issuance of an EAD is processed by the U.S. Citizenship and Immigration Services (USCIS). The process is described at their website: http://www.uscis.gov/green-card/green-card-processes-and-procedures/employment-authorization-document
Although the URL above includes the term green card, an EAD is not a green card. The green card is the card which shows a person is a legal permanent resident. If a person has a green card they don’t need an EAD. The two cards are quite different.
Form I-765 Application for Employment Authorization is used to apply. The instructions for completing form I-765 contains a complete list of eligible categories. It is found at: http://www.uscis.gov/sites/default/files/files/form/i-765instr.pdf
It is best for an interested individual to contact an immigration attorney, assuming the individual is not in the U.S. undocumented. However, if the person is a youth and undocumented, they may qualify to apply under the Deferred Action for for Childhood Arrivals (DACA) program. More information about DACA is at: http://www.us-immigration.com/deferred-action-application-I-821D.jsp