I would never recommend SendGrid. I took up their free account offer and never managed to send a single email - all got blocked - I spent days trying to resolve it. When I enquired why they got blocked, they told me that free accounts share an ip address and if any account abuses that ip by sending spam - then everyone on the shared ip address gets blocked - totally useless. Also if you use them - do not store your email key in a git public repository as anyone can read the key from there (using a crawler) and use your chargeable account to send bulk emails.

A free email service which I've been using reliably with an Azure website is to use my Gmail (Google mail) account. That account has an option for using it with applications - once you enable that, then email can be sent from your azure website. Pasting in sample send code as the port to use (587) is not obvious.

  public static void SendMail(MailMessage Message)
    {
        SmtpClient client = new SmtpClient();
        client.Host = EnvironmentSecret.Instance.SmtpHost; // smtp.googlemail.com
        client.Port = 587;
        client.UseDefaultCredentials = false;
        client.DeliveryMethod = SmtpDeliveryMethod.Network;
        client.EnableSsl = true;
        client.Credentials = new NetworkCredential(
            EnvironmentSecret.Instance.NetworkCredentialUserName,
            EnvironmentSecret.Instance.NetworkCredentialPassword);
        client.Send(Message);
    }

UPDATE: Google no longer let you use your google account name and password for sending e-mail. Instead you need to now switch on 2-factor authentication and create an app password and use that password in your app with your google account. Here is a step by step guide: youtube.com/watch?v=Y_u5KIeXiVI