In Short:
I have a PDF that customers can fill out. When they press the \"Submit\" button, I want to automatically send an email with the completed PDF
If your $_FILES array is empty, you probably are using wrong attribute value. Check if you have:
$_FILES
enctype="multipart/form-data"
This is required for form to allow sending files. Then you can use uploaded file and send it to the client, which seems that you are capable of.