Piece of SpaceX rocket believed to have crashed into the moon

The answer is no. Hubble's visible-light resolution is about 0.05 arcseconds, which translates to 90 metres per pixel, far too large to make out anything that's "merely" bus sized. An imager would need to in orbit around the moon to have any reasonable chance.

It seems like they're hoping to detect the plume of dust caused by the collision, however.

Multiple lunar orbiters will be passing overhead at some point. They will produce images of the after effects that can be used in a before/after comparison. But the crash will(was??) not be televised or live streamed which is probably more along what you are asking.

Astronaut 1: “Wait, the lunar landing system is just gravity?”

Astronaut 2: “Always has been.”