The official statement is that "despite VR in Godot being inspired by Cardboard, there is no support for Cardboard itself". However, I made a test project, and it ran on Cardboard.
Apparently, "no Cardboard support" means that it does not properly import all the configuration values from Cardboard API? Or does it? I could not tell if there is something still not working properly; the test project seemed to run just fine, albeit too slow for VR.