The web browser extension would communicate with the electron app server, NodeJS would not be needed in that scenario (the electron app includes the nodejs server code). You can write your web voice app with static client-side JavaScript which communicates with the Electron server through the browser extension.

Web Page <-> Bumblebee JS API <-> Bumblebee Extension <-> Bumblebee Electron App (DeepSpeech)

DeepSpeech with the pretrained english model is enormous (1.4GB) it's not feasible to load it into a web worker. It can run in a server, but then every website would have to run its own server side speech recognition servers which is difficult and expensive to scale.