Can Vex Apathetic only stun one unit per turn?

No, Vex, Apathetic is not limited to stunning only one unit per turn.

According to the card text, Vex's ability is a triggered ability that occurs "When an opponent plays a unit while I'm at a battlefield." Because this trigger condition can be met multiple times, and the card does not contain a "once per turn" restriction or limit, Vex's ability will trigger every time an opponent plays a unit, provided Vex is at a battlefield at that time.

As established in RiftJudge FAQ #9240 and #10242, each time an opponent plays a unit that satisfies the criteria (including tokens, if the effect that creates them uses the word "play"), Vex's ability will trigger and create a new instance of the stun effect on the chain, which will resolve individually.