Can I create an order form?

Is there a way to create an order form I can place on my website that will allow the customer to email the form to me once they've filled it out?

You can indeed create order forms with JotForm.  Here is our guide on the basics to get you started:

https://www.jotform.com/help/47-Order-Form-Basics/

You can send email notifications to your address as well as the user filling out the form. 

These are our guides on setting up email notifications.  The first is on setting up notifications sent to you:

http://www.jotform.com/help/25-Setting-Up-Email-Notifications

The second is how to setup notifications to your users:

http://www.jotform.com/help/26-Creating-a-Form-Autoresponder