Yes you should be able to run Vnc from a browser using the java viewer. I have done this in the past - when VNC was free and open source. However, it appears they have now gone to a paid model. You would have to make sure the school allowed remote access via the VNC port 5802 - their firewall might block access to it. I also found VNC to be relatively slow for screen updates. If the school district is using Windows (as many do), you could also use the Windows Remote Desktop client from the Netbook to access a Remote Windows machine set up as the server. You would need to install the Remote Desktop service on the remote machine, and enable it. If you had fixed IP you could access it via the IP, else you can use a dynamic DNS service to map it to a domain name and access it. Remote Desktop is very fast over broadband connections.
Thanks for the follow-up. I hadn't considered Windows Remote Desktop. That might work, if I can find a way to easily setup a remote machine. Never heard of "dynamic DNS". Sounds like I've got some more investigating to do...