Great question. You may not like the answer though.
Legally, the answer is yes you can under Article 51 of the Employment Services Act.
In practice, you may have problems. The local education authorities in some areas (Kaohsiung City) is one, require foreign teachers to have work pemits to teach English, and they will give you only if you are a native speaker from a list of approved countries. They are clearly iignoring the law, but it hasn’t been worth for anyone to challenge them.
In your case, I would show the school the law and your APRC and just go ahead and start working.