Man, tracking the release schedule for that novel can feel like a real scavenger hunt sometimes. I got into it years after the anime, so I was hungry for more of the original story. What I found is that the core novels by Kana Akatsuki, 'Violet Evergarden' and 'Ever After', are fully published and complete. That's your main story arc. So, if you're asking about new chapters for that specific series, the answer is no, it's not an ongoing serial with weekly updates. It's a finished work.
However, there is continuing online content, but it's more about side stories and specials. The official Japanese publisher, KA Esuma Bunko, occasionally releases short stories or bonus materials on their website or through special campaigns. These aren't new 'chapters' in the sense of advancing a main plot, but they are new pieces of the world. Translations for these trickle out slowly through fan efforts or official licensed releases, which Kodansha handles in English. Their digital editions sometimes bundle these extras. So the 'online update' scene is quieter, more about supplementary material than a continuous serial. My advice is to check the official Kodansha page for the ebook versions—if there's a new compilation, that's where it'll pop up first.