Rules clarified...
If they only came in for immunizations, and no history or exam was taken, no office visit should have been billed at all. You would especially not bill an office visit of 99213 with a routine diagnosis, like V700 or V202. V202 is routine child health exam, so technically, it can be used until age 18, but we typically don't use it past age 8, mainly due to payer definitions of well child care.
V70.3 is for a general medical exam for school admission.
You only bill an office visit when the key components are met during the visit. For a preventive (like 99394), there has to be an age appropriate comprehensive history and exam taken in addition to any counseling or immunizations.
For problem oriented, such as 99213, you must have specific elements of patient history (ROS, HPI, Chief Complaint, PFSH), and a physical exam, as well as medical decision making based on the presenting problem, diagnostic options, treatment options, and risks involved to qualify the visit for each E/M level. See CMS's documentation guidelines for further explanation.