I don't think that your computer being on the same LAN segment with others qualifies a problem. However, your network administrator may have tighter restrictions as a result.
I now assume the reason you get to the page via browser but not via PHP has to do with proxy server. In other words, to keep a tight grip on your school computers, you r network admin probably has your browsers configured to use an inhouse proxy server and then blocks access to port 80 out of your LAN. If that's the case, you might try and figure out how to send these requests through the same proxy server. Since I'm making an assumption here, I can't really help you further.
Good luck!