Get User for Audit Entries
Depending on how the application is setup, one (maybe the only?) way to pass data at the connection level would be to utilize the ApplicationName portion of the connection string.

You could have the application set this value based on the logged in user, making it accessible via APP_NAME() on the database side.

Example:

Server=myServer;User Id=myUsername;Password=Pass;ApplicationName="MyApp - User LowlyDBA"