How can I send a copy of the submitted form to my clients?

Why isn't it sending my client a copy of the form they just signed? Is there a way to set that up so they have the contract they signed on file as well?

In order to do this, you need to setup an autoresponder in your form.
You can see how to do this here: http://www.jotform.com/help/26-Creating-a-Form-Autoresponder 

Let us know if you've further questions.