At this time, yes, this plugin is designed to serve icons from a CDN, which assumes that the client and server are both on the internet and can reach both fontawesome.com
(for icon metadata) and either use.fontawesome.com
for the Free CDN or pro.fontawesome.com
for the Pro CDN.
Given the fact that other plugins and themes who load Font Awesome also seemed to tend toward loading from various CDNs, this seemed pretty normal to take that as a first approach.
Though now that you’ve raised the issue, I’ll grant that the old plugin that this new one replaces, shipped the Font Awesome 3 icon assets with the plugin code, and so loaded them locally (i.e. “self-hosting”).
Unfortunately, that approach isn’t very practical when it comes to version and dependency management. And it’s actually prohibited when it comes to Font Awesome Pro, since Pro licenses don’t permit distributing Font Awesome Pro in open source software like this plugin.
So yes, for now, I’d say that you should either try a different plugin, or, if you know you need to be able to work locally, I would recommend not using a plugin but just self-hosting.
To self-host: Download a recent version of Font Awesome 5 Free, unzip it and copy the assets you want to load (either the webfont/css or svg/js) into an appropriate folder in your WordPress or theme installation, and then perhaps in your theme’s functions.php
enqueue it. That will forfeit most of the advantages and features of this plugin, but it will get you the icons working, and working in a local environment.
Or, if you were satisfied with the previous 3.2.1
release of the old plugin here, you could download that older version to continue using in local.