(Forget this, keep reading to end) During any time alone with your new cadet, I am sure you could stage a photo op of an oath-taking and do the words between yourselves. It will mean just as much. The paperwork is all done, and the class will be officially sworn in as a group. That is essentially what it is at USNA, as parents do not see their plebe in uniform until they are released for a brief period after the group swear-in.
Now, I don’t know how USAFA I-Day works, so I don’t know if there is any window for a quick exchange.
As
@Academician notes, there doesn’t seem to be any guidance n the Appointee guide; you can google the Class of 2027 one.
Ah - a little more digging - belay my last - read this - you drop him off, USAFA owns him, no more contact:
www.usafa.edu