I'm half-way porting another turn of the century game engine to emscripten but I'm a little stuck on the networking so it's pretty cool being able to have practical examples to reference.
Multiplayer on the web is tricky. For non-action games you can get away with WebSockets but for arena shooters or other action games I think UDP is important, and you can only get that with WebRTC and all the baggage that comes with it. I'm using a library called HumbleNet to handle WebRTC, but I had to make a lot of changes for it to be usable. https://github.com/jdarpinian/HumbleNet
https://eikehein.com/stuff/sabatu/
I'll upstream the work to tr1x (which I contribute to) at some point.
WebRTC is available today in all browsers and supports peer-to-peer unreliable UDP.
The problem with WebRTC for this kind of application is the need for a whole separate "signaling server" just for connection establishment, plus STUN/TURN, plus your actual application server makes three required servers. It puts a really high minimum on the complexity of any WebRTC app. For a client-server only application you might be able to combine all three servers into the same binary. I haven't seen a library that does that, but maybe it exists, I haven't looked extensively.
The HumbleNet library that I use provides an implementation of the signaling server for connection establishment, and for the client app it hides all the WebRTC complexity behind the BSD sockets API. So all you need to do is host the server.