Can I store pre-wrapped shellfish in its store packaging?

Yes. In fact, it is recommended.  Food scientist Susan Brewer, Ph.D., explains why: "Every time you handle fish, it becomes contaminated by air, hands, and contact surfaces. Unless you plan to separate pieces of fish prior to freezing them, leave the fish in its original packaging.” Handling them is of less concern when freezing because microorganisms won't grow when the fish is frozen.
 
You can refrigerate shellfish in the paper supplied by the supermarket if it is tightly wrapped and taped up. However, if the fish is loosely wrapped, it should be re-wrapped. Because paper is somewhat permeable, it’s best to re-wrap the fish in plastic. Aluminum foil may leave a metallic taste. Before refrigerating, place the fish in a shallow pan--not on a plate--on the bottom shelf of your refrigerator to ensure  that drippings don't contaminate other products.
 
Fresh shellfish may be  pre-packaged using a process called modified atmosphere packaging (MAP). The air in the package is replaced with gases (such as nitrogen) that don’t interact, as oxygen does, with food components. Products wrapped this way may be puffed up on top because the air was displaced by gas that helps to preserve the product, and perhaps additional gas was added to be sure that all the oxygen was removed.  You should refrigerate the shellfish in this protective packaging.  
 
However, fresh shellfish should not be frozen in MAP packaging because water can migrate from the food to  the head space (the bubble)  created by the gas.  Also,  the plastic wrap over the head space can tear if something heavy is placed on it. Then, the product will dry out and develop freezer burn.  
 
Source(s):
Brewer, M.S. and Chapman-Novakofski, K. 2006.  Meat Safety for the Consumer, University of Illinois Extension
Susan Brewer, Ph.D.  University of Illinois, Department of Food Science and Human Nutritio