When thinking about antennas you have to keep two views in mind simultaneously:
When looking at the big picture, you have the magic of current flowing into an open piece of wire and radiating away. Maxwell's equations are needed to explain what's happening, and we know all sorts of approximations - especially near quarter and half wave wires.
But when looking at any small part of the antenna system, say less than $\lambda/10$ in size, then its just a regular electric circuit and Kirchoff's rules apply (of course just derived from Maxwell), but much simpler. So your DC or 60 Hz intuition is good enough for any small part of the circuit.
Your question doesn't say what frequency you're thinking of. If the frequency is high enough that the light bulb itself is $\lambda/4$ tall, say 1296 MHz and up, then it might work as an antenna and carry current. But if it's at 14 MHz, then the second case applies. You know a bulb won't light up if one side is connected to 120 V mains, the same applies here.