It's quite common that production sites are unable to run composer. Even if command line access is available the site code is read-only, so composer would be unable to make any changes. Either developers have to use composer locally and upload all modules and vendor files to the production environment, or a build system takes care of that task.
Your Drupal site will only have one
vendors folder and autoloader, which is used by Drupal and all modules; modules don't have to include the autoloader themselves.
I'm assuming you haven't used composer to set up the site initially, and have only downloaded the Drupal zip file and added your modules. Composer is unable to automatically determine dependencies from modules when using the zip file, but you can still add the dependencies yourself. From the root directory of your site (not the module with the dependency), add any required packages as a dependency to the project:
composer require ryanwinchester/hubspot-php
This will add the third-party package as a dependency of your site, install its files to the vendor directory, and also make it available to any modules.
Updating Drupal core with this setup is trickier however. You'll need to replace the
composer.json file with the updated version to make sure that you get the correct dependencies for Drupal core, and then re-add any of your additional dependencies.
You can read the documentation on using Composer to manage an entire site:
Using drupal-composer/drupal-project instead of the Drupal core zip file will allow you to add Drupal modules with composer, as well as automatically resolve and include any dependencies of those modules.