An HL7 interface will probably be the solution for you in this case. Direct access to data is generally not available or encouraged for most systems and RESTful or SOAP-based web services are essentially non-existent.
Most, if not all, EMR systems share data via HL7 interfaces. There are different interfaces to support different functionality through the hospital system. For example ADT messages contain admission, discharge and transfer information and SIU messages contain scheduling information. These interfaces are only made available on an as needed basis, so there will be no guarantee which interfaces are available from implementation-to-implementation.
Your best option will be to coordinate with the integration team at the hospital and get a listing of supported interfaces for their implementation.