Q:

How do I make a promise call to a third party server?

I am trying to make a call to extranet which should return me a parameter but the api doesn't await for it. 
Orders.prototype.add = function (data, db, callback, logger) {
    var extranet = new API_EXTRANET();
    extranet.addOrder(data, function(err, orders) {
        if (!err) {
            if(callback) callback(false, orders);
        }else{
            if(callback) callback(err, false);
        }
    }, logger);
};

I tried to return a Promise of it but nothing happens.
Orders.prototype.add = function (data, db, callback, logger) {
    var extranet = new API_EXTRANET();
    return new Promise((resolve, reject) => {
        extranet.addOrder(data, function(err, orders) {
            if (err){
                reject(err)
            } else {
                resolve(orders)
            }
        })
    })
};

Here is where I call the method:
order.addStoreOrder(order_data_update, db).then((response) => {
                                                            order.add(array_data, db, function (err, order_response) {

A:

You are mixing different way to handle asynchronous calls : callback and promises. You should stick to promises only.
// There is no need for callback here, the promise replace your callback
// Because you use of await I guess you should use it here as well
Orders.prototype.add = async (data) => {
  const extranet = new API_EXTRANET();

  return util.promisify(extranet.addOrder)(data);
};

// How to call it
try {
  const orders = await order.add(arrayData);
} catch (err) {
  // Handle the error
}

In case you want to wrap the addOrder return, you can do :
Orders.prototype.add = async(data) => {
  const extranet = new API_EXTRANET();

  try {
    const orders = await util.promisify(extranet.addOrder)(data);

    console.log(orders);

    return orders;
  } catch (err) {
    console.error(err);

    throw new Error('custom error');
  }
};

// How to call it
try {
  const orders = await order.add(arrayData);
} catch (err) {
  // Handle the error
}

Documentation: util.promisify()
Description: It turns a function that use callbacks into a Promise function

