Can eating store-bought cookies or cookie dough can illness?

It isn't likely, though opinions vary on the wisdom of consuming processed cookie dough in its raw state.  The primary concern is eggs, a common ingredient of processed cookie dough. Because eggs may harbor food-borne pathogens, they are pasteurized during the manufacture of dough. Nevertheless, conservative manufacturers such as Nestle and Pillsbury do not recommend consuming processed dough in its raw state. As Nestle explains, "As a food company, we take a very conservative approach to food handling and absolutely do NOT recommend consuming any raw product that is intended to be cooked." A June 2009 outbreak of E. coli linked to the consumption of raw, processed cookie dough appears to support this position, though it is unclear whether eggs were the primary culprit.  The FDA, on the other hand, has approved the use of pasteurized raw cookie dough for use in ice cream because it is "specially-produced cookie dough that's pasteurized."
 
Needless to say, raw cookie dough made at home with unpasteurized (shelled) eggs should not be eaten. Both processed cookies and cookie dough may contain gluten, a protein found in flour and other wheat products, which must be avoided by consumers who have celiac disease. However, more and more processors have begun manufacturing gluten-free cookies. The absence of gluten may be indicated on labels. Processed cookies and cookie dough  may also contain nuts, dairy products, and other common food allergens.
 
Moist or soft cookies are more prone to spoilage than dry cookies. (Spoiled cookies taste bad but are not dangerous to eat.) Many dry cookie varieties simply grow hard and become stale.  Cookies that contain nuts, butter or cream-based fillings or toppings may grow rancid. 
 
Source(s):
Nestle "Why You Should Not Eat Raw Cooking Dough"
Pillsbury "FAQs Refrigerated Cookie Dough"
Ochef.com "What Keeps Ruining My Cookies?"
FDA "Safe Eats--Dairy and Eggs"