If getting into your room early is an urgent need:
1. Do not list ANY requests on the reservation. Let them know that you will take whatever is available.
2. Use the on-line check-in. The check-in system asks for your arrival time and according to resort CMs they use that information to try and coordinate having a room available at the time you arrive.
Unless using on-line trumps all, honestly I would put your odds around 50/50 of getting into a room immediately. Sundays are tough because there are sooooo many people arriving. SSR may have 800-1200 rooms, but they also have HUNDREDS of families arriving in a single Sunday due to the weighted
DVC point charts.
IMO, the greatest chances of getting a room early come when you check-in very early. I've been in my DVC villa by 9am Sunday morning on multiple occasions. But as the day goes on, the flow of guests increases and new arrivals get bottlenecked behind housekeeping. That's when you end up having to wait until late afternoon to get into the room.
My kids are only a couple years removed from the nap ages so I remember it all too well. Whenever possible, don't put yourself in the position of relying on "Disney Magic" to make or break the start of your trip. Consider things like coming in a day early and staying off-site or at a cash resort. Then you're already in town to do that early DVC check-in. Adjust your arrival time accordingly (early or late...whatever works best.)
I always avoid making any big plans that first night. You wouldn't believe how many horror story trip reports invariably begin with the 5pm ADR that was disrupted by a late flight / late luggage / late room assignment, etc. Keep your first day VERY flexible and adjust accordingly.
Good luck!