Can eating applesauce make me sick?

It isn't likely. Applesauce is a cooked product. In fact, the FDA requires that all applesauce products be heat-processed--meaning pasteurized--either before or after sealing. This effectively destroys E. coli and salmonella, two pathogens associated with fresh apples. Don't bother checking for the word pasteurized on labels. Because non-pasteurized applesauce is barred from commerce, there's no need to indicate that applesauce on supermarket shelves has been pasteurized.
 
Applesauce doesn't readily support the growth of spoilage bacteria because its pH is in the acidic range. (High- and low-acid foods generally suppress the growth of spoilage bacteria.) As a result, applesauce will develop mold long before bacteria can multiply to a level that could make you ill. Once mold appears, applesauce should be discarded. Mold that develops mycotoxins can cause serious illness.
 
Source(s):
Motts.com "Frequently Asked Questions"
Health Canada "Novel Food Information"
Allexperts.com "Refrigerated Applesauce"