Using the WebGL API, how can I get a value from the depth buffer, or in any other way determine 3D coordinates from screen coordinates (i.e. to find a location clicked on),
From section 5.13.12 of the WebGL specification it seems you cannot directly read the depth buffer, so maybe Markus' suggestion is the best way to do it, although you might not neccessarily need an FBO for this.
But if you want to do something like picking, there are other methods for it. Just browse SO, as it has been asked very often.