Hi, Carl! This is an interesting question, because I've stayed up all night in the past to test this out for myself.
I usually find that I can start making my reservations around 6:00 AM. I have gotten up as early as 3:00 AM, and actually once or twice was able to make a dining reservation at that hour, but I would have done myself a much better service to just sleep until I could make more of the reservations.
Some reservations, you will still need to phone in for, including any dinner shows you'd like to do, or even the tea at the Grand Floridian. For the majority of them, however, you can begin online reservations right around 6:00 o'clock.
Thanks so much for the question!