Example of how to implement audio recorder in your Xamarin.Forms applications. - http://julianocustodio.com/gravar-audio

MainActivity.cs

using System;

using Android.App;
using Android.Content.PM;
using Android.Runtime;
using Android.OS;
using Android.Support.V4.Content;
using Android.Support.V4.App;
using Android;

namespace AudioDemo.Droid
{
    [Activity(Label = "AudioDemo", Icon = "@mipmap/icon", Theme = "@style/MainTheme", MainLauncher = true, ConfigurationChanges = ConfigChanges.ScreenSize | ConfigChanges.Orientation)]
    public class MainActivity : global::Xamarin.Forms.Platform.Android.FormsAppCompatActivity
    {
        protected override void OnCreate(Bundle savedInstanceState)
        {
            TabLayoutResource = Resource.Layout.Tabbar;
            ToolbarResource = Resource.Layout.Toolbar;

            base.OnCreate(savedInstanceState);

            Xamarin.Essentials.Platform.Init(this, savedInstanceState);
            global::Xamarin.Forms.Forms.Init(this, savedInstanceState);

            //Verifica se tem permissão
            if (ContextCompat.CheckSelfPermission(this, Manifest.Permission.RecordAudio) != Permission.Granted)
            {
                ActivityCompat.RequestPermissions(this, new String[] { Manifest.Permission.RecordAudio }, 1);
            }


            LoadApplication(new App());
        }
        public override void OnRequestPermissionsResult(int requestCode, string[] permissions, [GeneratedEnum] Android.Content.PM.Permission[] grantResults)
        {
            Xamarin.Essentials.Platform.OnRequestPermissionsResult(requestCode, permissions, grantResults);

            base.OnRequestPermissionsResult(requestCode, permissions, grantResults);
        }
    }
}