Embedding fillable forms on a website

The only real way of doing it is with the Adobe Embed API.

If you just embed the PDF file in your web-page it will be up to the browser to decide what to do with it.

It might display it inline using its own PDF plugin, or it might download it, or download and then open in a local application, etc.

I cannot recommend you use Acrobat (PDF) forms for this. So many things can go wrong, and losing even one form could be a disaster for you. You need a custom solution written by a professional web programmer, based on HTML forms, with a highly secure back end meeting all the security requirements.