Fixes: http://forums.xamarin.com/discussion/comment/12118/#Comment_12118

MyWb.cs

using System;

using Android.App;
using Android.Content;
using Android.Runtime;
using Android.Views;
using Android.Widget;
using Android.OS;
using Android.Webkit;

namespace MyWb
{
	[Activity(Label = "MyWb", MainLauncher = true, Icon = "@drawable/icon")]
	public class MyWb : Activity
	{
		int count = 1;

		IValueCallback mUploadMessage;
		private static int FILECHOOSER_RESULTCODE = 1;

		protected override void OnCreate(Bundle bundle)
		{
			base.OnCreate(bundle);
			
			// Set our view from the "main" layout resource
			SetContentView(Resource.Layout.Main);
			
			// Get our button from the layout resource,
			// and attach an event to it
			Button button = FindViewById<Button>(Resource.Id.MyButton);
			
			button.Click += delegate { button.Text = string.Format("{0} clicks!", count++); };

			var chrome = new FileChooserWebChromeClient((uploadMsg, acceptType, capture) => {
				mUploadMessage = uploadMsg;
				var i = new Intent(Intent.ActionGetContent);
				i.AddCategory(Intent.CategoryOpenable);
				i.SetType("image/*");
				StartActivityForResult(Intent.CreateChooser(i, "File Chooser"), FILECHOOSER_RESULTCODE);				
			});
			
			var wv2 = this.FindViewById<WebView>(Resource.Id.webView1);
			wv2.SetWebViewClient(new WebViewClient());
			wv2.SetWebChromeClient(chrome);
			wv2.Settings.JavaScriptEnabled = true;
			wv2.LoadUrl("http://www.script-tutorials.com/demos/199/index.html");
		}

		protected override void OnActivityResult (int requestCode, Result resultCode, Intent intent)
		{
			if (requestCode==FILECHOOSER_RESULTCODE) {
				if (null == mUploadMessage)
					return;
				Java.Lang.Object result = intent == null || resultCode != Result.Ok
					? null
					: intent.Data;
				mUploadMessage.OnReceiveValue(result);
				mUploadMessage = null;
			}
		}
	}
	
	partial class FileChooserWebChromeClient : WebChromeClient
	{
		Action<IValueCallback, Java.Lang.String, Java.Lang.String> callback;

		public FileChooserWebChromeClient (Action<IValueCallback, Java.Lang.String, Java.Lang.String> callback)
		{
			this.callback = callback;
		}

		//For Android 4.1
		[Java.Interop.Export]
		public void openFileChooser(IValueCallback uploadMsg, Java.Lang.String acceptType, Java.Lang.String capture)
		{
			callback (uploadMsg, acceptType, capture);
		}
	}
}

Is there any chance that this is still working 4 years later?
At least for me it does not.