The "recommended method" is to push your laravel application using the provided .gitignore file (edit this file to allow tracking of the composer.lock file though). This essentially pushes anything potentially custom to your git repo, but not the entire working application code (i.e. it basically leaves out the vendor folder)

Then on your production server you would pull the code down, and run "composer install" to download the dependencies (i.e. vendor folder). Then you are good to go. The whole idea being that you let composer manage the dependency files (3rd party libraries, such as the laravel framework itself).

So in essence, yes you need composer on your production server (and composer will download laravel + other dependencies in the vendor folder).

However, composer is just a helper utility. You don't NEED it at all. If you wish to continue without composer, you can git push EVERYTHING including the vendor folder to git, and deploy that to production without requiring composer on the production server. Although this isn't the recommended method, more of a "last resort" method.