I'm not sure if this is possible in Angular 2, but,
I want to know if there is any way/approach/support for developing embeddable widgets which can be integrated into other websites that perform certain functionality.
For example, a small piece of source code that generates a button on the 3rd party website. Clicking on this button generates some popup windows performing some actions contacting our web server.
Please share your thoughts on this.
Have you heard of Angular Elements? It was introduced in Angular 6. The scenario that you're referring to sounds like a perfect use case for Angular Elements.
Angular elements are Angular components packaged as custom elements, a web standard for defining new HTML elements in a framework-agnostic way.
So just think of them as self-contained widgets, that will play quite nicely with any of your web apps built on any framework/library.
Have a look at this enlightening NGCONF Talk by Rob Wormald on Angular Elements. You can find several tutorials on creating an Angular Element. Here's another one by Jeff Delaney from AngularFirebase.
Angular's Official Website also has a very nice and easy to understand example that you can have a look at here.
Hope this helps.