How to send an email via Arduino without an email account?
You may be able to do this by running the SMTP transaction yourself. You don't actually need a valid from address (though I probably shouldn't be saying that). There are a number of resouces for how to do this, here's a quick list:


Arduino Playground Email Client page
Adafruit
How to send an email with Arduino and a WiFi shield?


If you read an article telling you how to do SMTP in a generic sense (not using a particular provider's infrastructure) and it looks like they want a "real" e-mail address, you can try with a fake one. Chances are decent that it may work.

There are some fine points that may cause you grief if you go for the most simple of implementation (for example, just because my e-mail address is foo@bar.org doesn't mean that bar.org will accept an SMTP connection, the "right" way to do it is to check DNS for an MX record for bar.org).

It would be best to do (and log) any errors that you encounter. There are plenty of things that can go wrong, so tracking where it blows up will help you – or anybody who's trying to help you debug.

If you don't want to roll your own SMTPClient and FullIP look worth checking out.