I've been toying around with something similar myself, only I want push-to-talk from my phone. There's a route there with a WebRTC SPA, and it feels like it should be doable just by stringing together the right bits of various tech demos, but just understanding how to string everything together is more effort than it should be if you're not familiar with the tech.
What's really annoying is Whisper's latency. It's not really designed for this sort of streaming use-case, they're only masking its unsuitability here by throwing (comparatively) ludicrous compute at it.