The Medical Contact Card tasks is related to a Medical Report and the Student Medical Profile Security for Staff task allows Teachers to view Student Medical Information.
If you want teachers to be able to view Student medical information, you need to enable the Difference between Medical Contact Card and Student Medical Profile Security for Staff tasks for the Teacher role. The Medical Contact Card task will allow teachers to run a report from their roster that shows all of the Student's medical information. However, the report will pull up blank if the Teacher role does not already have the Student Medical Profile Security for Staff task.
Both of these tasks can be enabled in Core:
Select Users/Access > Profile
Select Manage Roles
Click on the Teacher role
Mark Student Medical Profile Security for Staff and Medical Contact Card