document.cookie is a web API. It is the way that a script in a web page can access cookies. As such, you wouldn’t normally use it in an addon. It is also generally a pain to work with since it consists of one great big string containing all the cookies “associated with” a particular document.
There are special APIs to return any or all cookies that you might be interested in, and then to work with them. It is for you to decide if they are “on” the current web page. If you are specifically interested in cookies sent in http responses, you could intercept the respons and read the headers, but you wouldn’t normally need to do that.
Cookie management APIs here (and associated pages, see links at the end):
https://developer.mozilla.org/en-US/docs/Mozilla/Tech/XPCOM/Reference/Interface/nsICookieManager2
WebExtensions API (coming soon to a Firefox near you):
https://developer.mozilla.org/en-US/Add-ons/WebExtensions/API/cookies