But it calls for a tip120. I can't get tip120's at my local electronics store although they do stock TIP31's and TIP122's. Would it be possibal to use one of these as a substitute?
Use a TIP122 and instead of the 2k2 resistor use a 150ohm resistor to get good saturation at that high current. You'll almost certainly need a large heatsink (given the Vsat information in the datasheet).
You might be better off with a logic-level MOSFET with Rds(on) < 0.04 ohms, should run cooler.
Yes, a darlington transistor is a poor choice for this application (and most applications) because of the heat it will dissipate. Use a logic level mosfet instead.