Can I still play my units on a battlefield I control if the enemy has Vex Apathetic on their controlled battlefield?

Yes, you can still play your units onto a battlefield you control even if your opponent has Vex, Apathetic on a battlefield they control.

However, be aware of the following interaction:

• Vex's Trigger: Because Vex, Apathetic has the ability, "When an opponent plays a unit while I'm at a battlefield, [Stun] it. They can't move it this turn," your unit will be [Stunned] as it enters play, and it will be restricted from moving for the remainder of the turn.
• No Targeting: Per RiftJudge FAQ #9312, Vex, Apathetic does not target. It programmatically selects the unit you just played. This means that even if your unit has protection or abilities that prevent it from being targeted, it will still be affected by Vex's ability.
• Chain Timing: When you play the unit, Vex’s ability triggers and is placed on the chain. Because the chain is active (Closed State), you cannot play [Action] spells to respond to this trigger (see RiftJudge FAQ #9475).

In summary, your unit will successfully enter the battlefield you control, but it will immediately be [Stunned] and unable to move for the rest of the turn due to Vex's ability.