This post, this one, and this one should all offer information and examples relevant to the task at hand.
Bottom line, one needs to account for the thermal capacity of the dry air itself, any moisture in the air that has to be condensed out and frozen, cooling any hardware that comes along for the ride, thermal leakage of the system, and match that against the capabilities of available cooling devices.
The ask here is likely within reason for a single-stage peltier-based system, and under no circumstance would I ever recommend a series multi-stage peltier system if there is any practical alternative; the things are wretchedly, awfully, despicably inefficient as a refrigeration mechanism and grow exponentially more so when one starts stacking the things up in series.