Thank you for the advice.
After the search it was already clear to me that three.js does not support the microphone access in the core. Beside the given links I had found some more
(some in German only Musik und Javascript = Aufnehmen, Frequenzanalyse, Visualisierung, WAV, MP3 | Thomas Eses
https://www2.ak.tu-berlin.de/~akgroup/ak_pub/abschlussarbeiten/2014/WernerErik_MasA.pdf
Beispielcode für den Mikrofonzugriff über HTML5 ) .
However, not the two very good ones from the answer. Thanks for that. 
I was just hoping that someone had already taken this path: access the microphone and then bring in three.js. (with a basic example)
My friend wants to use this in a local network only. He’s also a music teacher.
Addendum
I was able to upload the example http://webaudiodemos.appspot.com/AudioRecorder/index.html to my server. It’s working. Now I can include three.js.
It only results in WAV. Still looking for mp3.