You must apply for a work permit to work in Iceland if you come from a country outside EU/EES. You are allowed to work up to 22,5 hours a week except when working is a part of your studies or takes place during a study break according to the curriculum.
A work permit is not issued until after you get your residence permit. You apply for a work permit at the Directorate of Labour.
You may not start working before a work permit has been granted.