techism This very forum is a Flarum install running on Opalstack on a managed account so I'm certain you can run it on your own account.
Docker is 100% not required. I've never used the "flarum-ez-install" thing that you linked to, but it seems quite old and isn't really needed on a flexible hosting platform like Opalstack.
Instructions for installing composer are here: https://docs.opalstack.com/topic-guides/php/#installing-composer
After you've installed composer, create a MariaDB user and MariaDB database: https://docs.opalstack.com/user-guide/mariadb-databases/
Next go back to your shell user where you've installed composer and create your Flarum project. I'll use "myforum" as the project name in this example:
mkdir ~/myforum
cd ~/myforum
composer create-project flarum/flarum .
Back in the dashboard, create a Nginx (or Apache) PHP Symlink app pointing at /home/shelluser/myforum/public/: https://docs.opalstack.com/topic-guides/symlinks/
Once the app is created, route it to a site: https://docs.opalstack.com/user-guide/sites/
Finally, visit the site and Flarum should guide you through the rest of the setup.
Hope this helps 🙂