Can I remove LinkMachine from one site and free up the site license to be used on another site?

Yes. First uninstall LinkMachine from the site where you will no longer be using it, as described here. Then log in to a different LinkMachine installation that uses that same account, go to the Site Manager, and click on the X in the Remove column for the LinkMachine installation that you have uninstalled. Doing this will free up the site license that was used by that installation so that it can now be used with a new installation.