Web application are run by the ASP .NET worker process which runs under the machinename\ASPNET user. Did you give this account access to SQL Server?

Don't.

Create an explicit SQL Server login and user and use SQL Server authentication.

The other option is to use .NET impersonation feature. There's a setting in the web.config that tells the worker process to impersonate another account. I don't remember what the option is.

-Mike