I've taken all Empire held planets, including Coruscant, I've got Palpatine prisoner, but Darth Vader? Still cannot find him anywhere. The game's been going on stalemate like this for nearly "1000 days" now! All that effort and I can't get past 99.999% complete!

Any help?

I'd have to say it sounds like a bug. Even if Vader was sent on a mission to a planet on the other side of the galaxy, once it succeeded or failed he'd have to return to either the planet/fleet he launched from or the nearest Imperial planet. Too, any remaining imperial fleets jumping from neutral sector to neutral sector with him onboard would suffer maintenance shortfalls constantly, destroying the fleet bit by bit.

A 1000 days? Either the Empire has a MASSIVE fleet still out there moving around, or you got some kind of ultra-rare bug.

He is on a transport somewhere, this has happened to me once.
For some reason he was on a transport at Hoth.
Go through the maps and see if you can locate an imperial fleet of any kind and send your fleet to investigate.

I mean he DID say he was in a stalemate for a 1000 days - he took all imperial planets at well. So in theory maintenance shortfalls should have destroyed any vessels the Empire had left long ago (unless it was a long campaign and the Empire had hundreds of ships before you took all their planets).

It happens a lot that characters are hopping systems in random ships in the game (especially endgame when the limited AI is at a loss as to what to do next). One particular session I played was like yours: Vader was bumming around on a Lancer frigate or something just randomly hopping through some neutral systems. I basically had to blockade the entire sector with interdictors just to catch him once I knew what was happening.

But, you know, that was in July this year. I'm pretty sure the OP quit long ago in frustration.