The fact that you can download some libraries and not others is puzzling as I have personally never experienced problems using the Contributions Manager.
All the libraries available in the Contributions Manager are either provided by the Processing foundation (e.g. Sound) or third party developers (nearly all the other libraries). Each developer provides a stable URL where the Contributions Manager can download the library from so different libraries can be from different host servers. I don’t know if that will cause an issue with your network. 
Is it possible to try installing one of them away from the school network?
You might try installing one or more of my libraries because I know that all the links work, you can choose any from G4P, Shapes3D, Sprites, AI for Games, Steganos, QScript and Jasmine, they are good ones to try.
Also make sure they are using the latest version of Processing 3.5.4
If anyone else has any ideas please chip in, this is not my specialist area and I am probably missing something obvious.