You need to amend the permissions by editing the role to allow subscribers or contributors to upload files
Something like the below may work for a contributor, edit for subscribers too:
add_action('init', 'wp_allow_contributor_uploads');
function wp_allow_contributor_uploads() {
if ( current_user_can('contributor') && !current_user_can('upload_files') ){
$contributor = get_role('contributor');
$contributor->add_cap('upload_files');
}
}