@Egidius, when creating an XMLHttpRequest, you should use
var xhr = new XMLHttpRequest({mozSystem: true});
What is mozSystem?
mozSystem Boolean: Setting this flag to true allows making cross-site connections without requiring the server to opt-in using CORS. Requires setting mozAnon: true, i.e. this can't be combined with sending cookies or other user credentials. This only works in privileged (reviewed) apps; it does not work on arbitrary webpages loaded in Firefox.
Changes to your Manifest
On your manifest, do not forget to include this line on your permissions:
"permissions": {
"systemXHR" : {},
}