Is it possible to create an AcroButton that, when clicked, will create a form field on the page?
Yes, this is in fact one of the most common operations done with AcroButtons. Several of the included button actions add fields to a PDF Document, from adding date and filename fields, to placing Previous/Next page buttons, and even a sample that adds a field with custom appearance parameters like border style and font. Each of these button actions can be modified for your own purpose or used as a template for creating your own button action script.