@amrugg They already have some generic sounds for moving pieces/capturing etc.
I think if they were to implement different sounds for various moves, it might get a bit bombarded with sounds and confusing to some players. Others may think differently though.
I think at some point we might offer something like this, but I am not sure when.
Awesome suggestion!
@jdcannon If chess.com ever implements this feature I would suggest having an option to either enable or disable it? Not all players may want the sounds playing all the time, especially in a more fast pace game like Bullet, or a tournament, it may be seen as a distraction for some players. I just wanted to point that out, in case no one thinks of it.
And to be clear, I'm not against the feature whatsoever, on the contrary, I think it was a clever suggestion made by @amrugg! 
(This shouldn't be too hard, but if it is tell me
) In Settings, have an option to upload sounds. For Moving, Capturing, Castling, Promoting, Checking, and Checkmating. What do you think?