If you are already in Taiwan with a Visitor Visa for employment purposes (such as business, investment, employment-seeking), you do not have to leave the country to apply for a Resident Visa. You can apply to turn your Visitor Visa into a Resident Work Visa at the Bureau of Consular Affairs in Taiwan.
You must apply for the Taiwan Resident (Work) Visa after your employer obtains your work permit, and at least 8 days before your Visitor Visa expires.
Once the BOCA issues your work visa, you must apply for a Taiwan residence permit at a local service center of the National Immigration Agency within 15 days.