When you define Integrated Security=True in the connection string, any user who is currently logged in will try to connect to your database. When you run this application in the console or Winforms, this is your user account.
However, if you run it as a Windows NT service, this is the service account that the service runs on - in your case obviuosly NT AUTHORITY\ANONYMOUS LOGON .
And the error says clearly: this user account does not have permission to connect to SQL Server.
You have several options:
stop the NT service and change the service account to be the one who has access to SQL Server
allow NT AUTHORITY\ANONYMOUS LOGON register on your SQL Server and use your database
create a specific user (for example, an application user) in your SQL Server and change the connection string to use this user:
connectionString="Data Source=10.10.2.102;Initial Catalog=DataBaseName; user id=Your-Application-User-here;password=The-Secret-Password"
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