You can check in as soon as you get there. However, having a room ready is the issue.
If you have made absolutely no requests for your room (building, floor, close to food court, etc.) your chances have just increased. What I would do is call now and make one request. That your room be available as early as possible due to driving all night with children.
No room requests are guaranteed. But, it will be noticed and hopefully granted. If not, you can ask at check in for the next room available.
To do this, you have to be very flexible on your room.
But - I would prepare for the worst. There are lounge chairs by the pool so one of you can nap. The kids can either swim (don't know the time of year you are going) or take a long walk with the other adult. Music is right next to Sports and then it's Movies. Explore Sports first and then go to the other ones. If you are in the mood for a walk and the kids are hungry, take the up to McDonald's. There is a sidewalk the entire way. They have a big playground there. Maybe have one parent do the McDonald's thing and the second do the walk through the resorts.
And - in about twenty years, remind the kids of this. Lord it over them.