No, it won't work.
Cooling a system is all about moving heat. Heat is measured, for instance, in BTU (British Thermal Units). THE best way to move heat is to blow air. A simple fan is the BEST way to remove the most heat from an area. If you have issues with dust, you need more filtration to keep the dust out.
An eBay peltier refrigerator has barely enough BTU to cool a can of soda after 30 minutes or even more. There is NO way it will be able to keep cool anything that generates heat - NO way. That's like asking if you could take the radiator out of your car, and use a tiny USB refrigerator instead. Heck no!
Forget about active cooling like peltiers (especially) or air conditioners - they just won't move enough BTU unless they are VERY large units - VERY large! A large machine easily generate a higher cooling need than a whole family home. If you have to, use a shop-vac to blow air in to the cabinet - it will move a LOT of air (and with that air, heat). You can then use the high flowrate and cheap filter in a shopvac to keep the cabinet clean. Depending on your environment you may need to change (or clean) the filter after a certain number of running hours, or days, etc. Depending on how often you run the machine, you may need a high duty cycle motor or you will burn it out. Normal shop vac's probably are not meant to run for hours and hours and hours on end...
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