Any live moon stream?
Since the moon isn't visible all the time from any single camera location on Earth, such a website would have to be an aggregate of many different cameras around the world, displaying a single camera based on current time and weather conditions. Even in this scenario, there's a chance all the cameras will be obscured by clouds and there will be no view of the moon.

Such a website doesn't exist that I know of. Perhaps it would be an interesting project if you're interested in such a thing.