Can you tell us what the errors are?
The web site will typically run under the limited access 'NETWORK SERVICE' account. It's very likely that this account won't have the right access to SQL Server. I have never connected to MSSQL from a web app with anything other than a SQL Server login.
What is your motivation for not wanting a username/password in the web.config file? If the concern is someone seeing the security information, there are ways to encrypt sections of config files. I haven't done this myself so I can't offer any specifics.
-Peter
peterlanoie.blog