If Akali, Silent is played while Vex, Apathetic is at a battlefield, does Akali, Silent get stunned?

Yes — Akali, Silent is stunned by Vex, Apathetic.

Vex's trigger ("When an opponent plays a unit while I'm at a battlefield, [Stun] it") applies to the unit identified by its own trigger condition — the "it" is programmatically selected, not chosen or targeted (Rule 355.10.d). Akali's "I can't be chosen by enemy spells and abilities unless I'm in combat" only blocks effects that actually choose/target her, so it doesn't help here; the same protection wording on Baron Nashor likewise fails against Vex (FAQ #8685). It doesn't matter where Akali is played or whether combat has started — Vex's ability never enters "choosing" territory. Akali enters the board stunned and gains the "can't move it this turn" restriction, and that stun lasts until the next Ending Step cleanup (CR 424.1.a.2).