simulate this circuit – Schematic created using CircuitLab
Figure 1. OP's circuit redrawn.
If we redraw the circuit to lay it out in a conventional fashion (reading left to right with positive supply at top it becomes a little more obvious what is going on.
- If the switch is open, will the caps still draw charge/current?
It should be clear now that there will be an initial current to charge up the capacitors. As they charge the current will fall reaching zero when fully charged. Closing the switch will not affect the capacitor current.
2, Will this only happen when the load draws current from/through them? (Is the load required for the caps to charge?)
With the redrawn schematic you can see that the capacitors will charge whether the load is connected or not.
Note 500 kΩ on a 9 V supply will limit the LED current to \$ \frac {9}{500k} = 0.018 \ \text {mA} \$ which isn't enough to light an LED. 1 kΩ would give about 7 mA to split between the two LEDs but usually we would use one resistor per LED to balance the currents or connect the two LEDs in series to reduce the current required by the whole circuit.