So, here's the plan: Call 1-407-939-3463 at 7 AM Eastern, as advised. Actually, start calling a couple of minutes before, and keep hanging up and redialing until the lines open.
You may even want to do a 'test' run, any day during business hours (7 AM - 10 PM) so you can familiarize yourself with the voice prompts - you don't have to wait until the automated system finishes each prompt, you can respond as soon as each one starts.
There is, or was, a specific number to press for V&A reservations; the new system likely has the separate prompt. There may be additional, distinct prompts.
As soon as a CM answers the phone, say "Chef's Table" and the date. If you get it, great. If not, in the meantime make a reservation for the dining room.
Repeat daily, and if still unsuccessful, try intermittently before your trip.