There are two options and the one is a bit easy in terms of coding and the other will require a bit of coding and technique. First option is to try to be in contact with the IT technician for that hospital I'm sure he has access to web hosting files or should I say credentials of the web hosting which you can have a look at and not just anything but be more to the data base side and a background info on how the data is retrieved shared and linked. With that you will be able to create an application that will use the web database to retrieve data or
Second option, you will have to know the controls used on the web you are referring to so that you will be able to get data from it. Take a look at these examples:
The first one, Another one, And check web scrapping too.