Please forgive me, I am a hack. This isn't really my realm but I am making do.
Currently, I have an arduino Uno running an Adafruit 16 channel pwm servo board that lights up LED's for testing. Eventually the LED's need to be replaced with dc motors. I have the servo board simply sending pwm signals to transistors on a breadboard that are controlling the LED's. Only one at a time is lights up, so only one motor will run at a time. I might run two at a time in the future. My question is what kind of power supply do I need to run the motors?
I have read about problems with servo boards being underpowered because all of the servos require some amount of current all off the time but I am wondering if that is less of an issue because I am just sending the pwm to turn on the transistor.
You need a power supply that produces enough volts and current to meet the needs of the motors. You also have to take into account the voltage that will be used by the transistor, which can be anything between very little, up to about 2 volts, depending on the type of transistor you are using. Or, if using some kind of H-bridge, the voltage drop of two transistors.
This sounds barely doable with your current choice of components, but a great deal depends on the motor load.
The TIP120 transistors have Vce(sat) = 2 volts at 3 amps collector current, and Vce(sat) = 4 volts at 5 amps collector current. This is how much of the power supply voltage will be lost as heat, depending on the motor current. Plus, you have to make sure the transistors are saturated at the intended motor load, that is, supply enough base current.
When the motor is operating at 7.2 volts, the data sheet lists the stall current as 8.6 amperes and the current for maximum efficiency as 2 amps. Even if the motor load is light, the motor will draw close to the stall current when starting up. However, the TIP120 transistors won't conduct that much current so motor startup will be sluggish. You will need to put the transistors on heat sinks. If a motor ever stalls, the transistor will probably be destroyed.
Supposing you want the motors to get 6 volts at about 2 amps (maximum efficiency power load), you need an 8 volt power supply. However, you could use a higher voltage power supply and keep the PWM percentage less than full on. Your power supply must be able to continuously supply at the very least 2 amperes per motor, for as many as will be running simultaneously. If you want two or more motors to start up simultaneously, budget at least 6 amperes/motor.