Is there a way to distribute an addon internally from a local server
instead of publishing it to AMO (https://addons.mozilla.org/)
You can do this by specifying an updateURL in the manifest for your add-on.
The periodic check for updates can also be done manually by the end users.
Automatic Add-on Update Checking
Applications will periodically check for updates to installed add-ons by retrieving the updateURL. The information returned can be used to notify the user of an updated version to the add-on as well as inform the application of new application versions that the add-on is compatible with.
Source Extension Versioning, Update and Compatibility
updateURL
A link to a custom Update Manifest file that specifies available updates to the add-on. The format is described below. If enabled, the add-on manager periodically checks with this Manifest file to determine if newer versions are available. When not included, the add-on manager will still check for updates on AMO, using the ID of the extension.
...
Examples
<em:updateURL>http://www.foo.com/update.cgi?id=%ITEM_ID%&version=%ITEM_VERSION%</em:updateURL> <em:updateURL>http://www.foo.com/extension/windows.rdf</em:updateURL>
For add-ons hosted on addons.mozilla.org: You may not specify an updateURL property. By default, Mozilla applications using the Add-on Manager (such as Firefox and Thunderbird) will send update requests to addons.mozilla.org using the default web service. Every time you upload a new version of your add-on or change its compatibility parameters through the author interface, your update manifest will be generated automatically.
Format of the Update Manifest:
The Update Manifest is a RDF/XML datasource. For an example of an update manifest, see Extension Versioning, Update and Compatibility.
Source updateURL