What web browser are you using to text from your website? Many web browsers use their own PDF plug-in which is not made by Adobe. Many PDF readers cannot view all interactivity features.
I am using Chrome
I don't use Chrome but I recall that it uses its own PDF reader. Hopefully, a Chrome user can confirm that.
You'll run into that problem about PDF readers because users of your website will use a variety of different readers.
Why do a PDF form on a website? Just create an HTML form or even easier, use a google form.
You’ll spend way more time than it’s worth doing this as PDF.