If Vex is on a BF and a unit is played with Aurora at the end of turn, is it stunned the turn after?

Yes, the unit is stunned and will remain stunned into the next turn.

Here's the timing breakdown:

1. Ending Step (317.1) — Dazzling Aurora's end-of-turn trigger fires. The unit is played.
2. Vex, Apathetic triggers ("When an opponent plays a unit while I'm at a battlefield, [Stun] it") and resolves, stunning the unit.
3. Expiration Step (317.2) — All units heal, and "this turn" effects expire (Vex's "can't move it this turn" wears off).
4. Turn ends.

The key rule is 423.1.a.2: Stunned units lose the status "at the beginning of the next Ending Step." Since the stun was applied during this turn's Ending Step, the next Ending Step is the next turn's Ending Step. So the unit enters your opponent's turn still stunned, and can't contribute its Might to combat damage until the next turn's Ending Step begins.

Vex's "can't move it this turn" clause does expire during the Expiration Step, but the Stunned status itself persists — they are separate effects with different durations.