Change webext API calls to use browser instead of chrome

[eritbh]

Firefox implements a promise-based version of Chrome's chrome.* extension API, referenced as browser.*: https://developer.mozilla.org/en-US/docs/Mozilla/Add-ons/WebExtensions/API

It also provides a polyfill to support Chrome: https://github.com/mozilla/webextension-polyfill

Given we're converting lots of our code to promises, I think it would make sense to use a promise-based browser API since it doesn't seem like it's otherwise much different (shouldn't require much logic rewriting, just cleanup).

[creesch]

Yeah the problem with that used to be edge, with edge switching to chromium that hopefully shouldn't be an issue anymore.