[Frumph]

^ the user needs to have the correct rights for example

function add_theme_caps() {
    $role = get_role( 'contributor' );
    $role->add_cap( 'upload_files' ); 
}

add_action( 'admin_init', 'add_theme_caps');

If you want to add contributors and above to be able to upload_files, change that to subscriber if you want