Bit of a strange set up but have come across a project where Composer has been used in a local environment to get a project started. The original developer did not have ssh acce
You could copy the composer.json from the server to your local server, add the requirements and run composer update.
After that you can copy all files (composer.json, composer.lock and vendor folder to your server)...
Or you can copy local vendor/phpmailer into servers vendor folder, search for the entry of phpmailer in vendor/composer/autoload_psr4.php and add it to your servers vendor/composer/autoload_psr4.php.
Using this method also add phpmailers dependencies the same way.
composer depends phpmailer/phpmailer