I'm not sure what they are talking about: "handing out". There is no line. There is no place to line up. (except for a great cup of coffee at the Portal Store!)
It is self-service, there is no one managing the process.
In mid-August, when I and my associates were doing our research study, and stayed at that campground for two weeks, we did not see it full once. Must happen, but I think rare.
However, I'd say that anytime before, say 5pm, would guarantee that you'd nail a spot. Remember that a number of people move out at 2am, so there is plenty of space in the am.