I can give you my experiences:
Last August we stopped in at the Sandcastle club (BC) and the girl at the front suggested that you call a week or two in advance, but said that "unofficially", they can usually get you in the day of or the day before- as long as they have the staff scheduled. Stopping in Monday night to reserve Wednesday night gave them the time to add a staff member. Their ratios are pretty low- I don't remember the exact numbers but it's something like 1 staff for 10 kids, so if they have 11 kids scheduled, they can take 9 walk-ins without a problem. If the resort is packed and they are at capacity though you are stuck.
On the same trip we had kids night out come out- we called them Monday as well and we were #3 on their waiting list for Thursday night. They called us Wednesday afternoon with an available sitter.
I feel it was pure luck that got us kid's night out. That was during the week before labor day- when the parks are practically empty. That said, if you pick a night that isn't too busy, you will probably get it a few nights out. Who you have to compete with are the people who don't totally know what they are doing who may pick your night, and try to plan before they do. To be safe I would probably contact a kid's club 2 weeks out and Kid's night out maybe a month or more to be sure. That service was SO worth it. We went out for dinner and fireworks, when we came back the kids were in bed sleeping. They loved it. I think the cost for 2 kids was pretty close, but for 3 kids the sitter cost less and watched the one too young for the kid's club.