It’s not possible at the moment. There isn’t a web interface because we rely on the Ruby code in SketchUp to translate the model into a format that is recognized by VR Sketch. We decided against uploading the .skp file, because it would open a can of worms for us: we’d need some way to decode the .skp files on the cloud servers (or on the VR headset directly), and these machines are not running Windows nor Mac but Linux or Android—and there is no native Linux or Android way to do that.
We are focusing now on Mac support for the Oculus Quest. As far as we can tell, we should be ready in the next month or so to support at least upload-to-cloud, as you are asking for. Complete editing in SketchUp should follow soon afterwards.