If your child has chronic issues then I would check with his pediatrician to bring his/her medical records or at least a letter outlining what should be done for him/her in case of emergency. That said, although they are not close to WDW, there is Arnold Palmer Hospital for Children close to downtown as well as Florida Hospital for Children. You would probably want to head to Celebration Hospital first though and if they decide that a pediatric hospital would be the best place for your 5 year old then they will make arrangements to get you there. They are affiliated with Florida Hospital.
There is an urgent care location right at the exit of Downtown Disney off 535. If you need transportation it will be provided.