Can I go directly to the principal as a teacher in a private kindergarten?

Primary schools have directors and principals, teachers are responsible for teaching, and directors and principals are responsible for guiding the overall situation. Kindergarten is basically the director of kindergarten. So you should report to the director first and see which one the director asks you to take charge of.

If you don't want to go to the kindergarten to find someone to coordinate, then you can let the principal of the primary school communicate with the director.