Adding store bought ice directly into your wort will increase your risk of infection, as you are adding unsanitized ingredients to your beer.
Does this mean your beer will be guaranteed to get infected and taste bad? No. Some people add ice directly and have no problems. But you are increasing the potential for an infection, just as you would if you didn't sanitize your bucket, other equipment etc.
I recommend against this method, unless you make your own sanitized ice.
Primary - Pale Mild, Ginger Ale, Brown Porter
Secondary - Stupid Berliner Weiss that won't sour...