calibre can do that, assuming someone has previously made a news recipe to scrape the site in question.
Getting it to work reliably on any possible input sounds like an interesting challenge.
I don't really need such services myself, and I certainly don't know what options there might be, but kudos for trying and I wish you a lot of luck.
An observation: calibre can convert your generated EPUB to AZW3, alternatively you could use Amazon's
KindleGen (maybe in combination with
KindleUnpack to make it smaller).
Then you can simply download the AZW3 using the Kindle's builtin browser -- which can download Kindle-compatible ebook formats and automatically add them to your library.
You could also use calibre's codebase as a reference to implement your own MOBI/AZW3 conversion engine -- it is GPL3 and written in python. Or extract the bits you need (since installing a couple hundred MB worth of Qt and other support framework for a large GUI application is not ideal on a server that just needs to convert books to Kindle format).