If I move an empowered Akali Deadly Weapon alone into a battlefield, dealing 2 damage to a unit there, and then exhaust Akali legend to pull Akali to base, dealing another 2 damage to that unit, can I respond to the Akali Deadly Weapon's second "when i move" ability trigger and ambush in a Morgana Vindictive to that battlefield as the Akali Leaves, dealing 4 additional damage to that unit and potentially conquering that battlefield with the Morgana?

No — you can't Ambush Morgana to that battlefield in response to Akali's second "When I move" trigger.

By the time that trigger is on the Chain, Akali's move to base has already completed — Moves are instantaneous with no in-transit window to respond in (Rule 441.3), and triggers fire only after the inciting event is fully processed (Rule 383.2.c). So at the moment you'd want to play Morgana, you control no units at that battlefield. Ambush only permits playing to "a battlefield where you control Units," and that must hold through finalization (Rules 822.1.b, 822.3). Without Ambush's permission, a Reaction unit still can't be played to a contested/opponent-controlled battlefield (Rule 813.3.a) — and you don't control it, since the attacker never controls the destination mid-attack. So Morgana can't get there in response to the second trigger: no play-trigger damage, no Conquer off that line.

If you want Morgana in, the legal window is earlier: play her via Ambush in response to Rogue Assassin's activation while it's still on the Chain. Akali is still at that battlefield and in the Showdown then, so the Ambush location is valid. Resolution order: Morgana's "When you play me" trigger (deals damage equal to what's already marked — 2), Morgana enters, then Rogue Assassin resolves and Akali's pull-back trigger adds another 2. Whether you conquer afterward depends on the enemy unit actually dying from that accumulated damage — a Combat Showdown was initiated, so at its end Combat Cleanup heals all non-lethal damage (FAQ #11658); only lethal damage kills, and if the unit dies Morgana remaining there can establish control and Conquer (assuming you haven't already scored that battlefield this turn, Rule 447).