Login failed for user 'IIS APPPOOL\ASP.NET v4.0'

You can change the ApplicationPoolIdentity from IIS7 -> Application Pools -> Advanced Settings.

Under ApplicationPoolIdentity you will find local system. This will make your application run under NT AUTHORITY\SYSTEM, which is an existing login for the database by default.

Edit: Before applying this suggestion you should note and understand the security implications.

Looks like it's failing trying to open a connection to SQL Server.

You need to add a login to SQL Server for IIS APPPOOL\ASP.NET v4.0 and grant permissions to the database.

In SSMS, under the server, expand Security, then right click Logins and select "New Login...".

In the New Login dialog, enter the app pool as the login name and click "OK".

You can then right click the login for the app pool, select Properties and select "User Mapping". Check the appropriate database, and the appropriate roles. I think you could just select db_datareader and db_datawriter, but I think you would still need to grant permissions to execute stored procedures if you do that through EF. You can check the details for the roles here.