You can do an Online Application or apply at the Immigration Department of the Ministry of the Interior of the place of residence

1. If you are not in Taiwan, you should apply to the Taiwanese embassy
    
2. If you are in Taiwan, you should apply to the Immigration Department of the Ministry of the Interior Applicants who have entered the country with a visitor visa that meets the qualifications for changing schools must check the required documents and visa fee at least 8 working days before the expiration of the stay period, and apply to the Central Office for a change of residence visa.

Online residence visa application form
Q&A (Consular Bureau, Ministry of Foreign Affairs)