I sent an email yesterday not using my gmail setup but instead thru the internet site ( it was a contact us site for a hospital) and now I would like a copy of the email I sent to them. They have answered back but my email was not attached.
If you used a form on a web site to send an email there would not be a copy in your Gmail unless they also sent you a copy of your email when it was submitted.
Try checking your spam folder, but if it's not there it's probably not possible to get a copy of the message you submitted over the web.