I want to be able to embed a fillable pdf form on my web page to allow my users to fill out the form and then click the submit button to send it to my PHP script to be processed the fillable form in the backend on my server-side script PHP.
Is it possible? If so, how would I go about doing it?
You can embed the PDF file in a web-page but it's up to the browser to decide how to display it. It could display it inline (which is actually not a good idea because it would mean it would use the sub-par browser PDF plugin for that), or open it in Acrobat/Reader/Something else, or download it, etc. It will vary for each user.