I have multiple websites that need to send mail (after successful signup, etc.); I maintain a Postfix instance just for this (it never receives any incoming mail).
This Postfix instance is a pain: it is sometimes broken, is regularly attacked, produces enormous amounts of logs, etc. These problems could certainly be dealt with by an experienced Postfix admin, but that's not me.
Is there a service that would be just like Twilio, but for email? IE: a web service for sending mail.
Have a look at Jeff's recent blog post So You'd Like to Send Some Email (Through Code). In the comments are a few relevant links.
Evaluation of the different service providers is left as an exercise for the reader though.
You could make a very simple AppEngine app that would accept message details from POST, and send the message through AppEngine's Mail Service.
I think GMail might fit your description. You can connect using SMTP/SSL and send messages from your account using any application (you can also register your account to send from multiple addresses).
You can connect and send messages from PHP using PEAR Mail (many webapps use this already). Similar functions exist for other platforms.