How do you drive a high current load from a moderate current source?
Will a capacitor bank and high current relay do it?


Ignoring the inductive load for the moment since that raises an entirely different question, simply assuming that you need to supply 20A @ 12V for 5 seconds, the energy required is:

$$12V \cdot 20A \cdot 5s = 1200J$$

Let's do a quick calculation to get a feel for the size of capacitor bank to deliver this energy over 5 seconds.

Say that you have a capacitor bank charged to 12V and you wish to supply 1200J of energy.  Understand that, as the capacitor bank discharges, the voltage across must decrease.  So, for example, assume the capacitor voltage will decrease from 12V to 11V during the discharge.  The equation for the required capacitance is:

$$\dfrac{C}{2}(12^2 - 11^2)V^2 = 1200J \rightarrow C = 104F$$

That's an enormous capacitance but there are ultracapacitors available that fit the bill.

For example, two of these in parallel yields 116F @ 16V but it will set you back about $300.  I'm quite certain that you could find a power supply that can deliver the 20A @ 12V for considerably less than that.