FERPA is about educational records in the US. In the case of primary or secondary schools, school nurse visits may be educational records, but otherwise the relevant law is HIPAA. Under HIPAA rules, a doctor (for example) cannot divulge anything, even whether a person was seen, without the patient's permission.
It is not illegal to request such information from the doctor, but they are required to deny the request (if there isn't accompanying permission from the patient). You can always request information from the student, or request them to convey permission to the doctor (although the doctor probably will not personally deal with such requests, so this would go through the medical records department). If the service provider has been given permission to release information, there is no legal obligation on their part to actually do so, unless perhaps there is a direct request from the patient to provide a specific person with specific information.