There will be someone at the desk 99.99% of the time. Make sure they have your flight # on record. They watch the flight schedules of reserved passengers and maintain staffing and buses to transport reserved passengers to Disney. If your flight is late and they have your flight #, then there will be staff to meet you and transport you.
Now, having said that, I have read one report where a family got in very late, (like 3 in the morning, or something like that, and there was a sign at the desk stating that they needed to call xxxxx phone # to arrange their transportation).
That's why I said 99.99% of the time.
There may be a time when only one or two passengers are "left" on their list and those flights are hopelessly delayed to some unknown time, in which case they may go ahead and shut down operations and put out the sign. But I have only read of it happening one time on this group since the service began.
If the sign is out front, call the number and you will be taken to your resort by taxi.
Also, if you tag your bags with the yellow tags and have a flight that is that late, make sure you have stuff to go to bed and get up the next morning. Your bags will be delivered to your room the next day.