I always reluctant to say no when things are potentially doable but here the list of permutations is as long as your arm.
In the simplest possible case (an LED) it can almost be done like that but a) you will need a big cap, essentially into supercap territory - LEDs used for practical illumination use a lot more power than indicators b) it would need to be wired across both the LED and the current limiter and c) you would need to understand the control circuit to ensure you are not reverse biasing anything that might object to it. Then you have the practical problem of identifying an internal void to accommodate it, this isn't going to be negligible in size.
Don't want to sound patronising, it really isn't my intention, but in my view this falls firmly into the "if you need to ask, don't bother" category.
I'd experiment with being light fingered on the trigger instead, on most drills I find the light switches on with even the lightest touch, well before the motor starts up.