APIs information according to the US CBP are to be sent anytime between 72 hours before the flight to 30 minutes before closing the aircraft doors for departure.
Just remember there are many passengers who fly last minute (last minute as in last check-in time which is usually ~1 hour before the departure) and their information is also required to be sent to APIs, that's how they do it, they send the information in batches as they are available until 30 minutes before the flight not only for newly checked in passengers but also for passengers who changed their information at some point after the initial submission.
In addition to the above, I have travelled to the US at least two times with no reservation and just by showing up at the check-in counter last minute.
Source: CBP's APIS Frequently Asked Questions for airlines.