Spectrabloom LED light box
Question is - do I have to have two power supplies?

No, you can use one power supply for both. But you need to make sure, that the LED current is not flowing through the Arduino, since that can kill the Arduino. To do that, you need to connect the power supply to the LED driver and the Arduino in parallel. For example, lets assume you have bought a 5V wallwart/power supply, which is can provide enough current for your whole circuit (and some headroom for further development). Then you can connect the positive lead to the 5V pin of the Arduino and the Vin+ (positive supply voltage) pin of the LED driver. Also connect the negative lead of the power supply to ground on Arduino and Vin- (negative supply voltage) of the LED driver. That way the current for the LEDs comes directly from the power supply and doesn't need to flow through the Arduino, but past it.