We can't tell without more information but it sounds rather like you will need a [WiFi_SSID, WiFi_PASSWORD] pair for the ESP to connect to the WiFi, and a [Network_USERNAME, Network_USERPASSWORD] pair for their network to let you use the network (as if you were logging onto it by hand from a terminal).
If so, you'd provide the first pair as you do now, with the AT+CWJAP... command, to get your device connected to the WiFi. For the second pair, your Arduino will probably need to interact with a user logon screen to provide the second pair. You'll need to get those details from the school's IT department, and possibly mimic a terminal.