Install module that requires composer on server with only FTP
That error message is called from the modules install file.  It's checking to see if the PHPMailer class exists, and if not, let's you know the library needs to be installed before you can install the module.

If you successfully installed the module on your local dev site, then Composer downloaded the PHPMailer library to the root 'vendor' folder.  In the 'vendor' folder there should be a 'phpmailer' folder.  You just need to copy this folder over to your website's server (to it's root 'vender' folder) and then you should be able to install the module.