If vex Apathetic stuns a unit and then dies can i move my unit

No — Vex dying doesn't lift the restriction. Once Vex's trigger resolves and applies "They can't move it this turn," that effect is already on the unit with a set duration ("this turn"), not a "while Vex is on the board" effect. Vex leaving play has no impact on it.

The restriction (and the Stunned status) both expire during step 3d of the end-of-turn cleanup — that's when all "this turn" effects expire simultaneously (Rule 317.2.c; CR 424.1.a.2). So your unit stays Stunned and immovable by you for the rest of that turn, regardless of what happens to Vex.

One nuance: the "they" in Vex's text refers to the unit's controller, so it stops you from moving it. If your opponent (or another player) plays an effect that moves your unit, that's not affected — only moves made by the unit's controller are blocked.