The capacitor size you need depends on the amount of current being drawn, and the time it takes to switch (ie how long the capacitor needs to supply the power). If this is for a turn-on signal, it's likely both these values are quite small, and you don't need a very large capacitor.
Once you have rough values for time (s, seconds) and amps (A), the rest is some basic math*:
Caps are measured in farads (F). F = C/V. C = A * s. So F = (A * s) / V. Plug in your values and you get the minimum capacitance you need to supply the current you need. Real-world caps are not perfect though and have a voltage dropoff, so double or quadruple that number.
*disclaimer: IANAEE, I'm just whinging it, so I may have some errors.
Or, y'know, you can just use a 100uF cap and see if it works (which is what I'd do)
What exactly are you trying to do? If what you want is to turn on with ignition but then not turn off until you hit a switch, I can think of better ways to go about it.
Linear Mode
