A few questions first
1) Is the SQL server on the same machine?
2) If it's on a different machine is it in a domain?
3) Are you using a full SQL installation, or SQL Express?
I'm going to assume you're using a full version of SQL, because if you're using SQL Express user instances will not give you this problem.
If it's on the same machine then you need to add Network Service to the list of allowed users in SQL. To do this start SQL Management studio, expand the security folder, then right click on Logins and choose New Login. Click search on the "Login - New" dialog, making sure Windows Authentication is selected then enter Network Service in the object name box and click Ok. Choose the database you want it to access from the drop down list in the Login - new page and click ok. Finally expand out the databases folder and for each database you wish to grant access to expand out security, right click on users, then select the network service login name, give it a name in the dialog of Network Service and grant it the right access by checking the role in the Database role membership list. If you're messing around db_owner will be fine, you will want to lock this down later.
If the SQL server is on a different box and you are in a domain you can run the application pool as a domain user and grant that access on the SQL box. How to do this varies on the version of IIS.
Finally you can use SQL logins and passwords if the SQL server is configured to do this. Create a SQL user and then add them to the connection string like so
Data Source=myServerAddress;Initial Catalog=myDataBase;User Id=myUsername;Password=myPassword;