Refer.

Login failed for user 'NT AUTHORITY\ANONYMOUS LOGON' Possible Connection String Issue?

I added the line in my web.config

<identity impersonate="true" />
 as mentioned in the link but getting the following error

A network-related or instance-specific error occurred while establishing a connection to SQL Server. The server was not found or was not accessible. Verify that the instance name is correct and that SQL Server is configured to allow remote connections. (provider: SQL Network Interfaces, error: 50 - Local Database Runtime error occurred. Cannot create an automatic instance. See the Windows Application event log for error details. ) 

Refer.

SQL Network Interfaces, error: 50 - Local Database Runtime error occurred. Cannot create an automatic instance

System.Data.SqlClient.SqlException: An error has occurred while establishing a connection to the server. When connecting to SQL Server

An error has occurred while establishing a connection to the server. When connecting to SQL Server 2005 this failure may be caused by the fact that under the default settings SQL Server does not all

A network-related or instance-specific error occurred while establishing a connection to SQL Server.

A network-related or instance-specific error occurred when connecting to remote server in ASP.Net

Sql Server Error: A network-related or instance-specific error occurred while establishing a connection to SQL Server. The server was not found or was not accessible

NT AUTHORITY\SYSTEM does not have access to the database in SQL Server and hence first you need to give it permission from SQL Server.

But better would be to access SQL Server using a SQL Server user by specifying Username and Password.