The Find My App is the only place to see and track an AirTag in realtime. There is no other place for this
Location updates will depend on whether any Apple devices are around that can see the Airtag and update its location successfully to Apple's servers.
Schools would likely require all devices to be turned off, and as such may be hard for the AirTags location to update successfully.
Also note, AirTags traveling alone without the paired iPhone or linked devices will start to emit a sound to alert others of their presence and issue constant notifications to nearby Apple devices. This can be very disruptive and annoying to teachers and Staff, and you will very likely get an admonishment by school staff for doing this.
In short, AirTags are not designed to track people. Apple advises against doing this and when identified this is liekly to cause issues for you with the school.