No, you don't need to add anything to the config form. Once you have created the schema and cleared the cache the translation form is built automatically (if the Configuration Translation module is enabled) and you can access it via the Translate tab. It looks by the way completely different than the original form and contains only values which are translatable according to the schema.
As @Kevin commented Drupal expects more, to be able to connect the translation tab to a base route. For simple configuration, as in this case, define a mymodule.config_translation.yml file.
Example for /admin/config/system/site-information:
title: 'System information'
And a tab, so that the tabs are displayed when the second translation tab is generated: