Can I charm a unit that is played this turn but stunned by Vex Apathetic?

Yes, you can Charm a unit that's been stunned by Vex, Apathetic.

The key is who the "they" refers to in Vex's text: "They can't move it this turn." That restriction applies to the unit's controller (your opponent) — not to you. Charm is your spell moving an enemy unit, so the restriction doesn't stop you.

However, watch the timing window. Charm is an [Action] spell, not a [Reaction] — so you can't play it while Vex's trigger is on the chain in a Closed State. You'll need to wait until after Vex's trigger resolves and the game is back in an Open State during your Main Phase. Once there, go ahead — the stunned unit is a perfectly legal target for Charm, and you're free to move it wherever you like.