Can you send the client a link to the form and have them submit it? Where is it housed? The cloud? Or does it have to be on my website?

Every form has a direct URL. Its basically a link to us. We host the forms.

It is possible to customize the direct URL a little:

How to Create a Custom URL of a Form?

However, if you like you can also embed your form in your own website. 

Which Form Embed Code Should I Use?