They use to give some reservations to Disney and keep the rest for their own reservation system. Now Disney Dining handles all of it. You have to book online or call, just like you would at any other Disney restaurant.
We had Lunch reservations on our arrival day on Aug. 20th at 2:30pm, but our flight got changed to an hour late, so I needed to push back our lunch time. I looked online, and there were no other reservations available at all. So I called T-Rex directly using the Landry's Club option and got someone at the restaurant. He said he couldn't do anything and gave me the Disney dining phone number.
But! He gave me a great tip. Told me to cancel my dining reservation so I don't get a no-show charge. And T-Rex does take walk-up reservations. Just give my name when we get there, show our Landry's card, and we'll be seated within 15 minutes. That's exactly what we did, and we were seated in 5 minutes.