Is Ground plug replacement necessary?
The ground prong is important. If a failure occurs that connects the hot to the parts of the vacuum that are supposed to be grounded, a functional ground prong means the circuit breaker trips. Lack of it means you get to play with electrocution. Devices that do not have a ground prong from the factory are "double-insulated" meaning that two separate failures would have to occur to connect the hot to the case (and in most actual product, have non-conductive plastic casings so you cannot touch any metal parts)

The plug overheating is an additional reason to replace the plug and would be even if the ground prong was intact.

So, replace the plug.