It should work in every browser except Internet Explorer (Edge is fine), though I doubt that comes as a surprise. It's based on the 2.9 version of Winamp, if you're wondering why it doesn't look exactly how you remember it.
If you want to run your own version, the source code is available on GitHub. It's a work in progress, so don't expect it to do absolute everything the original did.
While Eldredge's isn't the first attempt at Winamp-in-a-browser, it's definitely the slickest and most up-to-date.