Electronic – How to a capacitor store charge whilst also passing current

It's frequently said that capacitors store charge. Just reading through Wikipedia, I find:

Daniel Gralath was the first to combine several jars in parallel into a "battery" to increase the charge storage capacity. Benjamin Franklin investigated the Leyden jar and came to the conclusion that the charge was stored on the glass, not in the water as others had assumed.

Because the conductors (or plates) are close together, the opposite charges on the conductors attract one another due to their electric fields, allowing the capacitor to store more charge for a given voltage than if the conductors were separated, giving the capacitor a large capacitance.

Here Q is the charge stored in the capacitor

Charge is measured in coulombs, and I know from the definition of capacitance that if a 1F capacitor has a voltage of 1V, then 1C of charge is stored in it. If a coulomb is 6.241×10¹⁸ electrons, then there should be 6.241×10¹⁸ electrons in this capacitor somewhere.

But now consider this. If I use a capacitor as a load to some AC voltage source, some current will flow (the precise amount depending on the voltage, frequency, and capacitance):

^{simulate this circuit – Schematic created using CircuitLab}

I know that current is flowing all the way around this circuit, because if I put a lightbulb on either side of the capacitor, it will light. But if the current is flowing around this circuit, how does the capacitor "store charge"? In other words, how can I ever put electrons into the capacitor if the current is flowing around the circuit, which means for all the electrons I put in the capacitor, the same number come out the other side? If I can't put electrons in without taking some out, then how can the capacitor be storing them?