As you know if you go to wordpress.org, you can download the latest version of Wordpress. You can unzip this, and upload those files to get started.
However, WordPress requires a MySQL database. So you will need access to a database before you can install Wordpress. This isn't something you can do over FTP.
If you already have the database information, then yes it's possible. But creating the database like that is not possible.
On a second thought, you may want to try to host WordPress database on another server, which won't ok for your clients since they will depend on that other server to function.
As long as the MySQL server allows the database to be accessed by external servers. When you set up WP, instead of "localhost", you'll put in the SQL server.
Instead of "localhost" (in the wp-config.php) you will put the IP address of the server and then the config.php will connect
// wp-config.php
define ( 'DB_NAME', 'yourdbname' );
define ( 'DB_USER', 'yourusername' );
define ( 'DB_PASSWORD', 'yourpassword' );
define ( 'DB_HOST', '111.222.0.1' ); // IP address of the server where MySQL is running
// make sure MySQL server will listen to the request from your WP server IP !
If the database and database user do not already exist, your only chance is to access the Linux shell, which you might be able to do via PHP, depending on the hosts security settings.
The PHP command is shell_exec()
IF you have the rights to do this, then you have to send the right series of commands to:
1) Create the database
2) Create the database user (with password)
3) Grant access of the database user to the database
They have done something similar here: https://stackoverflow.com/questions/33470753/create-mysql-database-and-user-in-bash-script
However, you won't have root access, and you don't even have cPanel account access, so it may not be possible.
Although you can create database with php code, an easier way to setup wordpress would be to use wp-cli and mysql commands. This link gives detailed tutorial on the same.
username,IP,passwordfor database server as well. Upload a DB admin script like Adminer or PHPMyAdmin to public_html & then try to access the database using the same FTP credentials. If it works, then create the database from there and provide the info. towp-config.phpto start WordPress installation.