Peltier is solid state, you can feed it intermittent power same as a led.
Get a logic level (referring to the gate, otherwise 10V is needed) MOSFET that can carry > 3A. Ebay had IRLZ44's cheap this summer, check out the datasheet, you get a few for $1.. I bought almost $10 worth at 14.5 cents each.
Bigger concern for you may be your code. If you don't block, you could be taking samples per millisecond.