No, you cannot work in the U.S. with an F-2 visa. F-2 visas are dependent visas issued to spouses or unmarried children of students who hold F-1 visas. These visas do not authorize employment in the U.S. Working in the U.S. requires a separate work visa.
To get a work permit in the U.S., known as an Employment Authorization Document (EAD), you must be an asylee, refugee, U visa holder, K-1 fiance visa holder, or spouse of a visa holder. However, DACA recipients and tourists with B visas are not eligible for a U.S. work permit.
It typically takes 90 days for non-asylum-based EAD applications to be processed by the U.S. Citizenship and Immigration Services (USCIS). For pending asylum applications, initial EADs can be processed in 30 days. However, delays can occur if the USCIS requests additional information.