@Toe_Save is right that a buxiban is a business entity and also right that OP has nothing to worry about.
That said, a buxiban, though not a “school” in the ordinary sense, is subject to the Supplementary Education Act and the education authorities (local education department and central Ministry of Education).
The type of buxiban that most foreigners have contact with is 短期補習班, literally short-term buxiban or short-term supplementary education class, commonly translated as cram school. This is the type of business entity referred to in the Employment Service Act Art. 46 Par. 1 Subpar. 4 and the related qualification standards that require either a degree or a teaching certificate (along with a health check, age 20+, and citizenship of a country whose official languages include the one to be taught). If OP did not have open work rights, the ESA would dictate the requirements for a work permit. As Toe and @tando explained, OP does not need a WP.
If you have any doubt about a buxiban’s category, you can ask to see the registration or (at least in some cities) look it up on the education department’s website.