This will work but you will have some issues with condensation, the burning of pretty much any fuel produces a fair amount of water vapor, which will condense on the cooling elements of your fridge (and on the meat inside) if the fridge is running while you put smoke into it.
This will gather creosote, tar, and other undesirable flavor elements. You can avoid this by running the smoke through a condenser before you pipe it into the fridge, or by using the fridge at ambient temps rather than running it while you smoke. Anything under 90F is workable for cold smoking meats and sausages, but for delicate items like cheese and fish you might need to wait for a MN winter and then do it.