Can I use Akali champion to trigger Mournfull Witnesses?

Yes. On your turn you can activate the Akali legend (Rogue Assassin) ability to move Mournful Witness out of an ongoing combat to your base, and that ends the combat — Mournful Witness's "when a combat that I was in ends" trigger fires and it becomes Empowered.

Key points:

• The trigger is a historical check (FAQ #11725): as long as Mournful Witness had already gained an Attacker or Defender designation in that combat, moving it to Base doesn't erase that it was in the combat. When the combat is unstaged during the Cleanup after the move (FAQ #11755), the trigger fires.
• It must be an actual Combat Showdown (opposing units present). If Mournful Witness is just in a Non-Combat Showdown over an empty battlefield, there's no combat to end, so no trigger.
• The Akali legend ability needs "it's your turn" and targets "a friendly unit in a showdown" — so Mournful Witness (or any friendly combatant whose departure ends the fight) works. If other friendly units remain at that battlefield, the combat doesn't end yet; the trigger will fire when the combat finally does end.