I don’t think there is any way to do that at the moment, though I could see that being possible (maybe) if you stored your notes in google drive, and the new tab could query a file in the vault, then display and maybe edit it. I doubt such a thing exists yet, but it would technically be possible. I would suggest moving this to #get-help or #knowledge-management , since any integration would just be linking a new tab page to wherever your vault is stored, so not including any obsidian plugin.
Building a custom new tab extension for Roam sounds like a cool project! If you’re interested in going that route, check out Chrome extension development guides. They’ll walk you through the process of creating a Chrome extension that could integrate with Roam’s web client and potentially provide the functionality you’re looking for.
While Obsidian doesn’t offer a direct web client, you can achieve a similar experience using third-party integrations or even by creating a custom web application. Exploring options like WebDAV servers or cloud storage integrations can provide web-based access to your Obsidian vault. If you’re comfortable with programming, building a custom web application tailored to your specific needs can offer maximum flexibility. For a more comprehensive guide on building Chrome extensions, consider referring to online resources that specialize in web development and extension creation.