Within Teamwork, you can prevent your users from uploading files directly to the projects on your site.
This setting is applied to all users on your site, up to and including site administrators and site owner.
Note: The setting for disabling direct file uploads is available on per-user Premium/Grow plans and above.
Click your profile avatar in the bottom left of your site and select Settings from the pop-out menu.

In the Settings area, go to the General subsection where you will see an option for direct file uploads. This setting is enabled by default.

If you want to prevent your users from uploading files directly to your projects, toggle the setting off (gray).

When disabled, users will no longer see the green Upload Files button within a project's Files tab.

Note: This setting only allows/prevents file uploads directly to your projects. Uploads via third party storage integration options will still be allowed even if direct file uploads is disabled.
Toggle the setting on (green) to allow your users to upload files directly to the projects on your site. When enabled, users will be able to upload files to projects without needing to use connected file integrations.

For more information, see: Uploading Files in the Files Area