How Do I Send An Email WITHOUT An Ethernet or WiFi Shield
If you mean plugged-in via USB, then no. There's nothing you can do on the Arduino that would let you use the computer's network connection directly.

You would need to have a program running on the computer which would communicate with the Arduino via serial. It would effectively act as an intermediary, sending the email on behalf of the Arduino.

Unfortunately, I suspect you'd have to write the program yourself. It's possible that something similar already exists, but at the very least you'd probably have to modify it to suit your needs.