The library for that seems to support their custom protocol as well as Dropbox and Google Drive these days.
There's also Solid which attempts to do something similar: https://solidproject.org/
EDIT: I missed that your question was specifically about browser access. tbh I think it makes the most sense to deliver the files directly by the server, with maybe some media controls for specific file types but even that is something the browser can basically take care of out-of-the-box.
Or do you mean websites automatically saving stuff to your cloud storage without any UI interaction. That sounds far too abuse prone.
I suppose you could sync locally stored preference data but this is pretty uncommon since you could just store the data with your account. Would just be adding more complexity to the browser for a feature that very few would use properly, and likely many would abuse.