Assuming the website can be browsed for free, you can use
calibre to download news from a variety of different websites -- or write your own recipe for a new website -- and email or sideload the created ebook to your ereader.
A number of people heavily use this feature, and have contributed a vast array of news recipes to calibre. I don't see any Tamil ones, but there are 24 different English (India) news sources. Hopefully something there will appeal.
You can also ask in the recipes forum if someone can create a specific recipe, or help you make it.