Can I get my own exposure notification app approved for the App Store? I have an app that reports the results of rapid SARS-CoV-2 nucleic acid tests and would like to automate the process of propagating the results of that test through the Apple/Google Exposure Notification API. Can I add exposure notifications to my app, or is it only possible for one app per region to get App Store approval?
According to Apple's Exposure Notification API Addendum to the Developer Program License, you need the Exposure Notification Entitlement Profile.
To get that entitelment, it states:

2.  Entitlement Request; License and Restrictions
2.1 In order to use the Exposure Notification APIs, You must be a government entity, such as a government health services organization, or a developer who has been endorsed and approved by a government entity to develop an Application on behalf of a government for COVID-19response efforts.

Unless you are part of an authorized government agency, you will not be able to get this entitlement or submit an app.