I've done this in the past. You can try it out at http://demo.audiopump.co:3000.
The only way to current get audio in the browser is with getUserMedia and the Web Audio API. This gets you PCM samples. Unfortunately there is no way to use the browser codecs programmatically currently.
I handle this by sending the PCM samples to the server. Now, I don't send them as 32-bit floats, which is the format they are in. I detect the buffer size for the web socket and adjust the bit depth accordingly. If there is enough bandwidth I will send 16-bit stereo. Otherwise I can drop down to 8-bit mono. On the server, the samples are converted back up to 32-bit float and encoded with Ffmpeg.
From there, the data is sent live to the SHOUTcast server.