Welcome to the wonderful world of runDisney, Brenda!
As a mother of three, I know all too well having to work out the logistics of participating in a runDisney event while traveling with family. Your best option is to seek out the services of the Disney-endorsed, in-room childcare provider, Kid's Nite Out. This is your best option as Disney itself does not have a childcare service that would be available at the time of day you'll be running. The resort hotel childcare centers are only open from 4:00pm-midnight for children aged 3-12. And when needing to board race transportation by 3:00am, you can just let them sleep.
Kid's Nite Out is an Orlando based company that offers certified childcare works that are available 24 hours a day, seven days a week (which works great for those early morning race times). For more information, select the highlighted link above. I also recommend booking with them early as race weekends tend to fill up fast.
I hope this helps, Brenda! Please do not hesitate to reach out to me with any additional questions you may have as you continue to plan and prepare to runDisney.
May the course be with you!
Faith