A work permit requires a BA or a combination of an Associate’s Degree and TEFL. That said, I have not personally seen any jobs listed that didn’t ask for a BA as a requirement.
Perhaps try looking outside of the major cities like Taipei. You also may want to check with the Taiwan Ministry of Education. Apparently some TEFL programs may not be acceptable by their standards. It’s unclear how they qualify them exactly so it is best to contact them directly.