It probably varies by state. Here in CT you must be 16 to be certified as an EMT. MRT's (first-seponders) can be certified at age 14. Many volunteer ambulance services will allow you to ride if you are under 18. Some have explorer or cadet programs specifically designed for high school kids intertested in EMS, fire, or police work.
To work in an ED or for a paid service you probably have to be 18. Some EMS services require their employs to be 21 due to restrictions by their driving insurance. Call around to places that you are interested in applying to and ask about their specific policies.