Wow. I don't even mind bumping a long-dead thread, this is an entertaining puzzle!
Here's the solution I came up with. I wasn't timing myself, but I'm guessing it took me about ten minutes. The position is plainly impossible, since the pawns couldn't get to where they are without any captures, but it does meet the specified requirements.
Enjoy!
Chess Sudoku


The position is plainly impossible, since the pawns couldn't get to where they are without any captures...
I'm too lazy to try to prove it, especially given my lousy attempts at a proof earlier in this thread, but I'd be astonished if this can be done with a legal position. I just spent a few minutes trying and didn't even get close.
This seems a lot cooler than a lot of newer threads, no trolls in sight.