Would the voice FAQ widget work for on any landing pages/websites such as Google business sites or Wix sites?

It won't be possible to install the Voice FAQ widget on Google's business site(https://business.site). But, you can definitely add the widget on Wix sites.

Why? - Because business.site does not provide you with the ability to customize the HTML/JS code of the website. It only lets you do cosmetic changes via an interface. On the other hand, WordPress or Wix has a full-fledged "Website builder" where you'd be able to inject custom JS scripts and markups. Sites created through sites.google.com however would let you inject the custom code.