You could try emailing the racetrack directly to see if they have anything from that far back. I'd guess that chances are pretty slim though. Up until about 5 years ago, the smaller tracks usually didn't archive anything but their biggest races.
I was searching for race replays of my mare's races to buy, and did extensive research and lots of emailing to the tracks she raced at, but they didn't have anything archived from more than 7 years ago. And my mare raced in 2003 and 2004.
Another option would be to try get in contact with old owners, trainers and the breeder if you can find out who it was online. Often times they have pictures and win photos, and many love hearing about horses they bred, trained or owned. Just another option if you're not able to find any videos. :)