Thanks! This is loading just the demo assets, but I got the Emscripten support merged into upstream ioquake3 so you can check out the code, build it yourself, and with the right game asset files you can play the full game, Team Arena, mods, or whatever.

https://github.com/ioquake/ioq3/pull/658

The only major feature missing from the original game is networking. Obviously kind of important for Quake 3. Should be possible with WebRTC DataChannel. And while the original game didn't have it, for the modern web a touchscreen control scheme is essential since half of everyone is on their phone. Of course it won't be great compared to mouse/keyboard but some mobile support is better than none. So those are the two features I'd like to add (besides performance optimizations; it's a bit slower than it should be for some reason).