Can a PDF fillable form post itself to an HTTPS URL?

Yes, there is an option (action) that allows pdfs to post themselves. The steps are as follows:

List item

1. Open Adobe Acrobat Pro

2. Select the button tool
   

3. Add submit form button

   

4. Add the url and how you want the form to be submitted. You’re done.
   

5. The warning shown when the user clicks the “submit a form” button.
   

This can also be done in Adobe LiveCycle.