If Vex stuns Teemo Strategist, does Teemo Strategist still do ability damage?

Yes, Teemo, Strategist will still deal its ability damage even if it is stunned.

According to Rule 423.1.b, being stunned only prevents a unit from contributing its might to combat damage during the Combat Damage Step. It does not disable triggered abilities, and it does not prevent a unit from dealing damage via a triggered ability.

Reasoning:

1. Stun Effect: Per Rule 423.1.b, "A Stunned Unit does not contribute its might to damage in the combat damage step." It does not prevent abilities from triggering or resolving.
2. Teemo's Ability: Teemo's "When I defend" ability is a triggered ability. As long as the ability has been placed on the chain, it will resolve as normal.
3. Interaction: Since the stun mechanic does not silence or negate abilities, Teemo will still perform the instruction to reveal cards and deal damage to the target.

As noted in the RiftJudge FAQ #3204, triggered abilities are separate entities from the unit itself once they are on the chain, and they will attempt to resolve to the best of their ability regardless of the status of the source unit (unless the source unit is removed from the battlefield and the ability requires the unit to be "here" to deal the damage). Being stunned does not remove the unit from the battlefield, so the ability will resolve fully.