The Arc browser has something like this with their Capture + Easels feature (example of this post: https://arc.net/e/A6D1FE0C-063A-4500-8D19-46581D16181E), and it supposedly does update.

Truth be told, I have no clue how the live-updating feature works when I share an Easel but it does capture the page as I see it

Edit: It does change on my local "Easel" almost instantly with my reply, but we'll see if that screenshot updates for others. A little experiment, I guess.