It's definitely possible, several extension (such as MetaMask) use those feature. The real question is, are there any contributor willing to add and maintain such feature?
The local storage is probably not a problem. Just the notes will not be synced between multiple devices.
Synchronization problem could be solved with cloud storage API. But it means more difficult implementation/maintenance (developer side) and longer initial setup (user side).
If OP really wants to save info into browser, maybe a Firefox extension like Very Simple Notepad could help, although I'd prefer a proper local text file.
Might as well install proper note taking application if OP bother install such extension since usually it has more feature.