In Munchkin, Can a Wizard use his charm ability when he has no cards to discard in the hand?
No he can't, although official evidence is a little circumstantial. This question is in an old version of the Munchkin FAQ (but not the current one):
Q. A wizard must discard his hand to use his charm ability. Can he do this if he has no cards?
A. No. If you have no cards, you don't have a hand.
This is corroborated on the Munchkin wiki:
You can't use the wizard's abilities if you don't have any cards in your hand.
As was already stated in the other answers, discarding an empty hand is not possible.
This clarification is missing in the new Munchkin FAQ, because the wizard cards' text changed in the newer editions. The changelog summarizes:
The Charm Spell now requires a hand of at least three cards. You cannot discard a smaller hand to activate this power. To balance this, the Flight Spell now may be used after the Run Away roll.
This means with the old rules you need at least one card, with the new rules at least three cards for the Charm spell.