My DD has a number of seizures over the years while we were at WDW. She doesn't require any medical care, just rest though. We have used First Aid in all the parks, but luckily haven't needed to use any emergency service (although I have seen them in action).
At the parks, if you are close to First Aid, you can go there. They are staffed by RNs and/or paramedics and will call for further help if needed. If not close to First Aid, have someone in your party approach any CM to report a medical emergency. Many of the CMs have 2 way radios or cell phones. If the CM you approach doesn't, they will know how to reach a supervisor who does very quickly. There are CMs at many attractions who are trained in basic First Aid and/or CPR.
As was already mentioned, ambulances/paramedica are close by. You don't hear sirens or see ambulances because the emergency responders come from off-stage. They can be there very quickly. We saw a child fall out of a stroller (one child was standing in the stroller and another tilted it, causing the standing child to fall). Before we had even figured out what had happened and that a child was hurt, CMs from First Aid had already responded with an emergency kit and a few minutes after that EMS arrived with a stretcher and paramedics.
Celebration Hospital is very close.
In your resort, you would call the front desk and let them know there is a medical emergency. They would call EMS to your room.