Code run automatically even web application not running

If you want your mail functionality independent of the application's state (whether it is running or not), then you should look at SQL server's Database Mail feature.

Configure your database mail feature (by adding proper profile and SMTP server/account). Navigate to Object Explorer-> Management->Database Mail, right click and select Configure Database Mail.
Once you are done with Setup, test out the same by doing right click and select "Send Test E-mail".
If you are all set on step 2, modify the trigger to use msdb.dbo.sp_send_dbmail to send email to appropriate user (or users).