I am in the template editor for my WP site, and I want to upload a new file, but I can't find that option anywhere. I want to make a new php file for a custom portion of the site. Where can I find this functionality?
You can install a File Manager within Wordpress, Search the Plugins.
Or the easyist way to be honest is simply login via FTP to your web host locate the template folder within wp-content/themes/yourtheme/ and create a file with the name you want to have it. i.e wp-content/themes/yourtheme/mycustom.php just insert in the empty file. and then your see it editable within the theme editor.
Hope this helps you out.