Once someone has completed and submitted a form, can I have the submition sent to an email?

Hello, that's how it should work by default when you create a form. A corresponding Email Notification will also be setup. Just in case you don't have one, here's how you can set it up: Setting-Up-Email-Notifications 

On the other hand, if you already have an Email Notification setup BUT you're not receiving any notifications on your email, please give us the Form ID/URL you're working on, along with the email address in question so we can have a closer look.