If I play Punch First with Vex in combat, does Yordle Explorer draw a card?

Yes, Yordle Explorer will trigger and you will draw a card.

According to the official FAQ regarding Yordle Explorer:

"Cards that check costs look at the printed value on the card, not the actual amount paid to play it. You don't have to pay the full cost for the trigger condition to be met."

Even though Vex, Cheerless reduces the cost of your spells by 1 Power while she is in combat (making Punch First cost 1 Power instead of 2), Yordle Explorer only cares about the printed Power cost on the card. Since Punch First has a printed Power cost of 2, it satisfies the "2 Power or more" requirement for the draw trigger.