Peltier coolers need quite high current at low volts.  The one I used can be driven up to 2V which will draw 2A.

Power supplies are complex things of which I have little understanding!  The way I am driving the Pelt is by using a LM317T variable regulator supplied from the 5V output of an old PC power supply.  The circuit looks like

I used 1K for r1 and variable 1K for r2.  An additional 400 ohms from output to earth allows the voltage to be adjusted without the Pelt attached.

Its quite compact.  Sadly the heatsink shown was not up to the job and a larger one was needed.