I'm not sure if this counts, but back in the late 1980s, there was a local UHF channel that played all cartoons, all the time. They didn't have much advertising (and probably didn't last more than a couple of years ... HMM, CONNECTION) but one of the things they used for time-filler between programs were these music videos using clips of Hanna-Barbera cartoons set to various popular songs -- the only one I remember for certain was for "Freeway of Love". Google tells me that these videos were called HBTV (Hanna-Barbera TV) and it's nice to know I'm not the only one who remembers them -- thank you Internet! They weren't fan-made, which is why I'm not sure if it counts, but aesthetically they were very vidLIKE, and I loved them; I'd always look forward to seeing them.

The first fan-made vids that I remember seeing were some AMVs that my sister obtained via Napster in about 1999 or 2000. These included ones for Magic Knights Rayearth, the Final Fantasy movies, and Ranma 1/2. The quality was decent, even by modern standards, though many of them had screen artifacts that made it clear they'd been either dubbed off VHS or used some VHS source.

Not too long after that, I started finding vids online, usually as an accidental side effect of looking for fic. Most were AMVs, though I also found a few for SG-1 and Buffy. I believe animemusicvideos.org started up around that time; it was my main source for vids for a long while.