How can I create a button that will trigger an e-mail of an entire form to a specific client?

Note that when a Notification is fired and there are multiple forms, you get to choose the form to use in the Notification.

If you do need a PDF, then that is not currently native functionality.  I have all my clients use the Juiced Exact forms plus tool for that and sometimes in combination with their courier tool.  http://www.quickbasetools.com/exact-forms-plus

They can help you with that or other QuickBase Solution Providers like myself can.