Is it possible to connect Webflow to a custom-built NodeJS backend? The user when clicking a button, would send a request to the server and the response that the server sends would be used for creating a custom UI widget.
Your best bet is to build your site in Webflow and then export the HTML. That way you can host it however you like, build any widgets you like, and have any server-side code you like.
A good working example of something similar (yet extended far beyond what you’ve mentioned) would be Memberstack. There are many services that sit as a “layer” on top of Webflow serving up custom content, etc… Another example would be Wized which is, at a very high-level, the same concept.
Here’s an example of (one way) I’ve leveraged that with Webflow to create a course site with a custom UI widget - checkout the “completed” feature: