Police believe the incident happened at Grove Road school between 5pm on Tuesday, September 4, and 10am on Wednesday, September 5.
One of the windows of the double decker bus, which has been parked up and converted into a library, was left shattered.
North Yorkshire Police said: "In particular, we are appealing for information about anyone seen in the school grounds between the hours stated."
Anyone with information that could assist the investigation should contact North Yorkshire Police on 101, select option 2, and ask for Brendon Frith.
You can also email [email protected]
If you wish to remain anonymous, you can pass information to Crimestoppers on 0800 555 111.
Please quote the North Yorkshire Police reference number 12180166119.