Give access to store manager

To give access to the Quick Export Plugin to your Store Manager (or any other role), you can use the wqep_caps filter, and return ability the user need to have to use the plugin. 

add_filter( 'wqep_caps', 'add_export_to_manager');
function add_export_to_manager($cap){
	return 'manage_woocommerce'; //or any other role
}

For the Store Manager, the related ability is  manage_woocommerce. If you want to find a specific role, please refer to the WordPress documentation : Roles and capabilities