How can I add my health care institution to the list of providers at https://support.apple.com/en-us/HT208647

Apple are not here; we are all end users like you with no extended knowledge of special programs like this.

I looked at some of the support pages and see no way for an end user to request an institution can be added. I believe babowa is totally correct in that your provider/institution must make the contact with Apple. I would start by talking to your provider and /or your providers medical records supervisor.

I would guess that you would need to get the administrator of that organization get in touch with Apple coporate.