My 5 Year Old Has Developed Eye Bags

This has to do with facial structure at the top of the cheeks.  It is very unlikely to be related to allergies, or sinus issues.  So-called allergic shiners are a myth.

Most Likely these are developmental changes related to your child facial growth.  They may change with time.  It is important to differentiate true eyelid prominence, from those below the eyelid (upper cheek).  That being said, everyone is different and if you have noticed a drastic change - I have found that mom's always know.  You should have him seen by the pediatrician who can tell you more, and suggest further evaluation if needed.  Best of luck

If your son's eye bags seem to have started abruptly, he should be evaluated by a physician.  While you say he's not allergic to anything, he may in fact have some low grade environmental allergy or even a sinus problem that is causing what you see around the eyes.