Do I need a heatsink for a MOSFET
The simple answer is that you really do need a heat sink.

Take a look at figures 1 and 2 of the data sheet. At a 4.5 volt gate voltage, a typical Vds at 5 amps is about 0.6 volts for a case temperature of 25 C. 0.6 volts times 5 amps is 3 watts. As Linkyyy pointed out, the nominal maximum thermal resistance is 63 degrees/watt, so this suggests a 190 degree rise, for a nominal temperature of 215 degrees C. If you now check figure 2, you'll see that at 175 degrees, for the same Vgs, Vds is now about 0.8 volts, and power is now 4 watts. This is not a good trend. For that matter, figure 9 establishes that the absolute maximum junction temperature is 175 C, so you know you're looking at trouble.

And all of this is predicated on using typical values. Trust me on this, if there is one thing you should learn it is never, ever, to design a circuit using typical values for critical functions. Always, always, always use worst-case. Murphy's Law reigns supreme here.

Granted, since this circuit isn't failing by an order of magnitude, or anything like that, you probably don't need all that much of a heat sink to stay safe. But you do need one.