So you want to programmatically fire off a form notification email?  Here's one way to do it.  Let's be clear that I have no idea if this is the "correct" way.  I just got sick of asking this question myself and seeing other people asking it, so I figured I'd try to solve it myself. Also, you'll need to figure out a way to populate the entry id (or entire entry) as well as the action_id.  See the comments in the code.

	$entry_id = 72; // somehow you have to set the entry id of the form entry you're dealing with
	$target_action_id = 177; // id of the action you want to fire off - you can get this from the database WP_POSTS with the post_type = 'frm_form_actions'
				 //  or get it from the FF UI by opening the action and scrolling all the way down to the bottom right where it says "Action ID:

	$entry = FrmEntry::getOne($entry_id, true);
	$form_id = $entry->form_id;
	$type = "email"; // whichever you want
       	$actionToDo = FrmFormAction::get_single_action_type( $target_action_id, $type );

	$form = FrmForm::getOne($form_id);

	FrmNotification::trigger_email( $actionToDo, $entry, $form );