Ooh! Ohh!! Pick Me I Know!!!
Yes you can do walk ups for breakfast at the Captain's Grille! I HAVE!!! You run the variable risk of being delayed or even shut out if it is insanely busy or they have not adequately staffed the restaurant, but if it were me, I would mosey on over at some point prior to my stay and ask the staff what they think about a walk up during the time frame I am considering.
If I knew for sure that we definitely wanted to eat here, like lets say a nice breakfast before our plane departs I would book a definite time so we wouldn't be delayed....
But if I was rolling out of bed all hung over and desperate for a lobster omelet, I would haul my sorry butt down there as soon as I could muster the strength and take my chances at a walk up. The mere sight of me in this condition would be enough for the staff at the restaurant to take pity on me and hide me in a dark corner of the restaurant, preferably not near the wavy moving picture, and rehydrate me with citrus filled electrolytes.
I'm thinking that this second scenario is probably closer to your actual circumstances so I'm sure my experience will be of assistance to you.
~NM