I'd like to put a Linux parallel to the gateway device which will record all web pages visited.

Not just HTTP gets, but actually capture the files off the wire and write them to disk. Something that will save images too and modify the html code to link to the locally saved image files.

I need to see what the user saw when they went there, and have it locally stored, rather than just having a link to some site which then updated or went offline.

How would I go about accomplishing this? I have full control over the network and the computers btw.

It can't actually save files, but it does buffer webpages: SQUID. We used it for our university before employing BOA (faster). It has the possibility to buffer webpages to a certain degree on the gateway and offer it to clients in order to reduce the workload on the actual connection. But I am not sure if this function can be used for logging purposes...