If you want to read the AppFabric Monitoring Database from a Web Application your web app will have to have permission to login.
If you try to run a web app that does not have permission you will likely see this
Cannot open database "AppFabricMonitoringDB" requested by the login. The login failed. Login failed for user 'IIS APPPOOL\DefaultAppPool'.
You have two choices.
Rather than directly granting logins via SQL, Windows Server AppFabric has security groups that you can use.
Open a Command Prompt with Administrator privileges and run the command
NET LOCALGROUP AS_Observers "IIS APPPOOL\DefaultAppPool" /ADD
You may need to restart SQL Server or Reboot your machine for the change to take effect.