project.json

{
  "version": "1.0.0-*",
  "buildOptions": {
    "debugType": "portable",
    "emitEntryPoint": true
  },
  "dependencies": {
    "WindowsAzure.Storage": "7.2.1"
  },
  "frameworks": {
    "netcoreapp1.1": {
      "dependencies": {
        "Microsoft.NETCore.App": {
          "type": "platform",
          "version": "1.1.0"
        }
      },
      "imports": [
        "dnxcore50",
        "portable-net45"
      ]
    }
  }
}

ProxyTest.cs

using System;
using System.IO;
using System.Net;
using Microsoft.WindowsAzure.Storage;

namespace ProxyTest
{
    public class ProxyTest
    {

        public static void Main(string[] args)
        {
            WebRequest.DefaultWebProxy = new MyProxy();
            HttpWebRequest.DefaultWebProxy = new MyProxy();

            var resp = HttpWebRequest.Create("https://www.google.org").GetResponseAsync().Result;
            var body = new StreamReader(resp.GetResponseStream()).ReadToEnd();
            Console.WriteLine(body); // works through proxy

            var connectionString = "DefaultEndpointsProtocol=https;AccountName=...;AccountKey=...";
            var account = CloudStorageAccount.Parse(connectionString);
            var blobClient = account.CreateCloudBlobClient();
            var container = blobClient.GetContainerReference("test");
            var blob = container.GetBlockBlobReference("test");

            Console.WriteLine(blob.DownloadTextAsync().Result); // does not work, ignores proxy
            Console.WriteLine("Done");
        }

        public class MyProxy : IWebProxy
        {
            public MyProxy()
            {
                this.ProxyUri = new Uri("http://proxy.mycompany:8080");
            }

            public Uri ProxyUri { get; set; }

            public ICredentials Credentials { get; set; }

            public Uri GetProxy(Uri destination)
            {
                return this.ProxyUri;
            }

            public bool IsBypassed(Uri host)
            {
                return false; /* Proxy all requests */
            }
        }
    }
}