Sample showing how to export a SQL Azure database to BACPAC format.

SQLAzureExportSample.cs

using System;
using System.IO;
using System.Net;
using System.Runtime.Serialization;
using System.Text;
using System.Xml;

namespace SqlAzureBackup
{
    class Program
    {
        /// <summary>
        /// Requests that SQL Azure exports a database to blob storage in BACPAC format.
        /// </summary>
        /// <returns>A GUID representing the job.</returns>
        static Guid Export(string serverName, string databaseName, string userName, string password, string blob, string key)
        {
            // Call the REST API, with an XML document containing the job details and credentials.

            // NB This API does not seem to be documented on MSDN and therefore could be subject to change.

            // NB It's a good idea to do this on a copy (see CREATE DATABASE AS COPY) for transactional integrity

            // North Central US              https://ch1prod-dacsvc.azure.com/DACWebService.svc
            // South Central US              https://sn1prod-dacsvc.azure.com/DACWebService.svc
            // North Europe                  https://db3prod-dacsvc.azure.com/DACWebService.svc
            // West Europe                   https://am1prod-dacsvc.azure.com/DACWebService.svc
            // East Asia                     https://hkgprod-dacsvc.azure.com/DACWebService.svc
            // Southeast Asia                https://sg1prod-dacsvc.azure.com/DACWebService.svc


            var request = WebRequest.Create("https://am1prod-dacsvc.azure.com/DACWebService.svc/Export");

            request.Method = "POST";
            
            Stream dataStream = request.GetRequestStream();

            string body = String.Format("<ExportInput xmlns=\"http://schemas.datacontract.org/2004/07/Microsoft.SqlServer.Management.Dac.ServiceTypes\" xmlns:i=\"http://www.w3.org/2001/XMLSchema-instance\"><BlobCredentials i:type=\"BlobStorageAccessKeyCredentials\"><Uri>{0}</Uri><StorageAccessKey>{1}</StorageAccessKey></BlobCredentials><ConnectionInfo><DatabaseName>{2}</DatabaseName><Password>{3}</Password><ServerName>{4}</ServerName><UserName>{5}</UserName></ConnectionInfo></ExportInput>", blob, key, databaseName, password, serverName, userName);

            System.Text.UTF8Encoding utf8 = new System.Text.UTF8Encoding();
            byte[] buffer = utf8.GetBytes(body);
            dataStream.Write(buffer, 0, buffer.Length);

            dataStream.Close();
            request.ContentType = "application/xml";
       
            // The HTTP response contains the job number, a Guid serialized as XML
            using (WebResponse response = request.GetResponse())
            {
                Encoding encoding = Encoding.GetEncoding(1252);
                using (var responseStream = new StreamReader(response.GetResponseStream(), encoding))
                {
                    using (XmlDictionaryReader reader = XmlDictionaryReader.CreateTextReader(responseStream.BaseStream, new XmlDictionaryReaderQuotas()))
                    {
                        DataContractSerializer serializer = new DataContractSerializer(typeof(Guid));
                        return (Guid)serializer.ReadObject(reader, true);

                    }
                }
            }

        }


        static void Main(string[] args)
        {
            Guid guid = Export("MYSEREVER.database.windows.net", "MYDATABASE", "MYUSERNAME", "MYPASSWORD", "https://MYACCOUNT.blob.core.windows.net/backups/newfile.bacpac", "MYKEY");

            Console.WriteLine(guid);
        }
    }
}

Thank you so much, this is exactly what we need! Only thing I would add is that you've left off 2 data center URLs, I believe you can see the complete list here