The error means the web application doesn't have access to your database. On Windows 7 / IIS 7, by default each application pool has its own user. It seems the idea is to improve security by restricting what that web application can do (in case it gets compromised and controlled from the outside). You can change what user the application pool is running under but that will defeat its own purpose. A better way seems to give the pool's user the needed permissions (and not a bit more).
On the SQL Management Studio connect to the server you want your web app to connect (tested with SQL server 2008). Go to
Security -> Log ins
right click, New Log in. In the form that comes up leave everything as default except username, where you have to type whatever username the web app is trying to use, in this case 'IIS APPPOOL\MyApplicationName'. Note that the search function of that dialog fails to find or check as valid that user, but nevertheless it works.
Still on the SQL Management Studio connected to the server go to
Databases -> *YOUR-DATABASE* -> Security -> Users
right click and New User. I'm not sure if the user name field there has any effect, I just set it the last part of the username, like MyApplicationName. Then I've set the login name to IIS APPPOOL\MyApplicationName. You can click on the ... button and use the check and search, this time it works. If you don't do the previous step, the user will not be present here. Then give it whatever permissions you want to this user, like db_datareader.
And that's it, you've given permission. If lack of permissions was your problem, then it should be solved (or at least, I've just solved it that way).
I have a total amount of 2hs of experience with IIS and about three weeks with SQL Server and less than two months with Microsoft technologies so take my advice with a grain of salt, I can be totally wrong. (If another person can confirm these are the right steps, feel free to remove the last warning).