Can it be done?
If so, any information on the subject is greatly appreciated. (If not with the Web Audio API, is there some other API that lets me achieve this?)
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.
Web Audio API
Here's a fiddle https://jsfiddle.net/bomzj/beap6n2g/