Even in the summer while busy there was availability. Different nights have different themes, so you may want to check out the schedule when you get there
I've not seen people turned away but I'd recommend booking to avoid any dissapointment. We generally book on the day at City Hall. The DCR one doesn't seem as busy as the Cheyenne or Santa Fe but I guess this is because you need a car to get there!
We ate there in Decembe and the food was very nice. Not as good as the Cheyenne imho though.