Web Audio API, get the output from the soundcard

Derek is correct. If You have a realtek card (I don't know about support from other manufacturers) You set it to stereo mix, and set the stereo mix to default microphone. You can then create a source node linked with your mic using navigator.getUserMedia.. Google and You Will find it.

I just made a fiddle for you: Audio from soundcard

EDIT: the link now goes to the right version of the jsfiddle (the other one had the bars wrong).

It's possible to capture global audio output with Web Audio API for audio cards that support Stereo Mix feature, otherwise you could use Virtual Audio Cable.

Here's a fiddle https://jsfiddle.net/bomzj/beap6n2g/