Would applying lidocaine to my skin make me not feel the hep b vaccine?

Little effect: Skin is a wonderful, protective barrier and will not allow Lidocaine to penetrate to block the discomfort from an injection of any sort. You could ice the injection site prior to needle insertion which can help. The discomfort is minimal however.

Not really: The needle must go deep to the skin into muscle where the vaccine material is dispersed. The stretch of the muscle fibers will hurt, regardless of any transient numbness on the skin surface.