Do capacitors conduct electricity while charging?
For DC. It is not correct to say that the capacitor conducts current. Current is the movement of charge. Charge moves through the resistor and is stored in the capacitor like a reservoir, as it fills up the pressure (Voltage) increases. When the capacitor voltage reaches the base turn on level the current flows into the transistor turning it on. 

In practice a current does move on the opposite side of the cap. This is to balance the charge that is building on the positive plate of the capacitor. It gives the appearance that the capacitor is "conducting" but the current does not flow through the capacitor because the dielectric between the plates is an insulator.