Do I need to use any external parts with a voltage regulator?
From the Datasheet (Figure 17, Standard Regulator Circuit):

2. Although no output capacitor is need for stability, it does improve transient response.
3. Required if regulator is locate an appreciable distance from power supply filter.

Both can be omitted if you arn't too concerned with how clean the regulation is. If you are using it for a cell phone charger, you would want the caps. If you have a load that switches between low and high current often, you would want the caps. Also depends on how dirty/noisy the ac adaptor is. But if all you are doing is adding a few leds with resistors and nothing else, you don't really need that clean of a line.
The input cap is to filter noise while the output is to keep the line steady during certain times. The &quot;appreciable distance&quot; is wire length, in less than 10 inches.
That said, you could get those caps from dead electronics. 0.1uf is common, and you could use the same size on the input cap as well.