I need to move a single plugin to a different site. It seems the plugin is custom made and is attached to a database. How do I find all the proper files and move it to the new location successfully? Is there a plugin that can do this or would I have to create a row in the database and manually copy and move the files via FTP?
Plugin file, if created properly, will create it's own db entries for settings etc if needed. You should just move the directory of the plugin from your wp_content/plugins/ directory from one installation to the new one.
You could also zip the plugin's directory (either by ssh, cpanel, or download via ftp and then zip on your computer) and then in the new install of wordpress dashboard go to the plugins/add-new and upload the zip.
If you're looking to export the post entries that have been created by the plugin you can either pull directly from post meta (search for post-type that the plugin created) or use an import/export tool, there are plenty of those plugins on wordpress repository.