Definitely. It’s referred to as self-hosting, where you put all of the assets on your own web server and host them yourself. This plugin does not support self-hosting. So if you want to self-host, you’d be doing that instead of using this plugin.
The exact steps you’d take to accomplish that will vary depending on your web server environment.
One way to get the assets needed for self-hosting is from the download .zip file available for each release. If you have a fontawesome.com account, the download would be available from your account page. There are other ways of getting the assets for Font Awesome Free, such as the GitHub repo.
One of the ways you could setup self-hosting on WordPress would be follow a pattern similar to these docs. Except that wherever you see URLs in the docs that refer to CDN locations, you’d use URLs that refer back to your own web server–to the locations where you’re self-hosting the Font Awesome assets.