Access webpage's localStorage from a Chrome extension
I just tested it, and if you access localStorage
from the context of a content script, you get the webpage's localStorage. So, nothing special is required besides injecting a content script into the webpage you need.
To communicate the value from the content script you can use Messaging API, and you can use chrome.storage.local
API to save data in a way that's accessible from both the content script and the background page.