From what I understand, you cannot continue without at least checking this checkbox.
When the order is validated, in the order details I can see under the memo section an “additional details” with the informations completed in the cart page looking like this :
3 days ago, I uploaded a new version of the theme online.
The main change was to change the type of cart from a “Page” type to a “Drawer” type (+ some CSS).
From 3 days ago, I identified 4 orders that don't contain the “additional information” part (I have a lot of other orders completely normal). I didn’t find the same behaviour even going back a month before so I believe it's releated to this upload.