Your form uses Content Controls. The method you are attempting to use is for legacy form fields. To do it with Content Controls takes more than using a button.
Here is Greg Maxey's page on the same Add-In with thoughts on what you need to do with Content Controls to do the extraction.
https://gregmaxey.com/word_tip_pages/extract_da...
Here is a page that will take you step-by-step through the process of writing a macro to do what you want.
https://stackoverflow.com/questions/29524537/ex...
These are links to pages on one or more Word MVP websites. Those pages contain accurate safe information that I think will help you. They were written by a Microsoft Word MVP. However, as an Independent Advisor I am required to give the following notice when providing non-Microsoft links:
Required Notice: This is a non-Microsoft website. The page appears to be providing accurate, safe information. Watch out for ads on the site that may advertise products frequently classified as a PUP (Potentially Unwanted Products). Thoroughly research any product advertised on the site before you decide to download and install it.
Personally, if it were me, I would talk with someone in IT and see how you can get the Add-In. Otherwise, you are going to be re-ineventing the wheel.
Alternatively, shift to using legacy form fields.