In Japan, we change out of our normal shoes and into “indoor shoes” at school. Many people just use crocs or sandals because they are cheap, but really you can wear any kind of shoe so long as it is a “school-only” pair. We recommend it be a slip-on style, as you will also have to change out of these shoes any time you use the bathroom (bathroom slippers are provided).
Open-toed shoes are perfectly acceptable at many schools. A good strategy is to have a pair of open-toed shoes or sandals for the summer to keep your feet cool, and closed-toe shoes for the winter to keep your toes warm. This is Miyazaki, yes, but people have gotten frostbite here. Protect your fingers and toes!
Furthermore, everyone takes their shoes off when entering the gym, be they indoor school shoes or regular outside shoes. Many teachers have a separate pair of indoor gym shoes. You can get by without having a pair at all; for ceremonies, etc., you don’t have to wear shoes. None of the students will be wearing them and several other teachers won’t be as well. However, in winter the gym floor will be freezing, and if you want to join in for class match or other sports activities in the gym, having a pair of indoor gym shoes would be a good plan. Unless you have large feet, you can probably pick up a cheap pair over here rather than taking up space in your luggage.