Distribution grouop does not get message from website

What mail service is the form on your website using? It probably sees the email as coming from an untrusted source and doesn't apply the rule.

Do you have the IP address(es)/hostname(s) where the message is coming from in your whitelist in Exchange and SPF record?

Dear Skyler Kincaid, thank you for your speedy response.

When you say "in whitelist", do you mean executing
      
$list = (Get-ContentFilterConfig).BypassedSenderDomains
$list.add("domain.com")
Set-ContentFilterConfig -BypassedSenderDomains $list


in Exchange PowerShell enough?

No because the messages are coming from your domain right? You will want to add the IP that is being used to send the messages.

Hello, could you post the rule configuration, and a mail header?

Dear Skyler,
No it is not.  It is coming from the website hosting company's server name.  Sending mail address looks like this.
anonymous@ip-xx-xx-xx-xx.ip.secureserver.net, where xx-xx-xx-xx is the IP of the server that hosts our website.  Thank you.

Dear Gianpiero,
Here is the header, but I do not know what you mean by rule configuration.    Thank you.

Received: from in3c.electric.net (72.35.12.43) by
 LMHF001.MyCompany.local (192.168.1.8) with Microsoft SMTP Server id
 14.3.248.2; Tue, 20 Dec 2016 13:34:15 -0500
Received: from 1cJPEs-0003J4-56 by in3c.electric.net with hostsite:432482
 (Exim 4.87)      (envelope-from <anonymous@ip-xx-xx-xx-xx.ip.secureserver.net>)      id
 1cJPEt-0003cM-3J      for info@MyCompany.com; Tue, 20 Dec 2016 10:34:15 -0800
Received: by emcmailer; Tue, 20 Dec 2016 10:34:15 -0800
Received: from p3plsmtps2ded01.prod.phx3.secureserver.net ([208.109.80.58])      by
 in3c.electric.net with esmtps (TLSv1.2:ECDHE-RSA-AES256-GCM-SHA384:256)      (Exim
 4.87)      (envelope-from <anonymous@ip-50-62-80-58.ip.secureserver.net>)      id
 1cJPEs-0003J4-56      for info@MyCompany.com; Tue, 20 Dec 2016 10:34:14 -0800
Received: from ip-50-62-80-58.ip.secureserver.net ([xx.xx.xx.xx])      by : HOSTING
 RELAY : with SMTP      id JPDnclPNSY0EoJPDncbSXI; Tue, 20 Dec 2016 11:33:07 -0700
x-originating-ip: xx.xx.xx.xx
Received: (qmail 28358 invoked by uid 10212); 20 Dec 2016 13:33:06 -0500
X-Additional-Header: /var/www/vhosts/MyCompany.com/httpdocs
To: <info@MyCompany.com>, <support@lfmedia.com>
Subject: New Message From  MyCompany Communications - Contact Us
X-PHP-Originating-Script: 10212:class-phpmailer.php
Date: Tue, 20 Dec 2016 18:33:06 +0000
From: "Y. Ballan" <mail@MyCompany.com>
Reply-To: "\"Y. Ballan\"" <yballan@MyCompany.com>
Message-ID: <809ffb33c514945a3ef000d3c48a985a@www.MyCompany.com>
X-Mailer: PHPMailer 5.2.14 (https://github.com/PHPMailer/PHPMailer)
MIME-Version: 1.0
Content-Type: text/plain; charset="UTF-8"
Content-Transfer-Encoding: 8bit
X-CMAE-Envelope: MS4wfIZQkOIzkuHE6e8XlfhDD72ORM4iwDYqoRBrrs2o9j33L8JJPYb4xH814cLQL90lcAXcYVt7OsqI7W2QB6UpwgYFg6d+n5+nBlEhUhQIhIEaQewtYg3Q
 OwL/+nRCDMtQPodSw2N8fAyMrXWwOjCobtxoSHnukuyPJkGrxOZ0Cz0qxu5PuGTmUNis3EnBeCRb6zV5tZEp0PE8ylurb5NRX0r1+8vc5kRMxdhfjgQIgPHL
X-FM-OS: Linux 3.x
X-FM-GeoIP: US
X-Proto: esmtps
X-Revdns: p3plsmtps2ded01.prod.phx3.secureserver.net
X-HELO: p3plsmtps2ded01.prod.phx3.secureserver.net
X-TLS: TLSv1.2:ECDHE-RSA-AES256-GCM-SHA384:256
X-Authenticated_ID:
X-Origin-IP: 208.109.80.58
X-Env-From: anonymous@ip-50-62-80-58.ip.secureserver.net
X-DKIM: none
X-PolicySMART: 5442992
X-SPAM-Status: NO, ALLOW RULE=MyCompany.com:CADF
X-Virus-Status: Scanned by VirusSMART (c)
X-Virus-Status: Scanned by VirusSMART (s)
Return-Path: anonymous@ip-50-62-80-58.ip.secureserver.net
X-MS-Exchange-Organization-AuthSource: LMHF001.MyCompanyCom.local
X-MS-Exchange-Organization-AuthAs: Anonymous

Dear Mr. Rossi,

I am sorry for not getting back to you sooner, our office has been closed and I have not accessed our server.
I will get the info as soon as I get in tomorrow.

Dear Mr. Rossi,

After I changed the recipients as you suggested, it started to work.  Thank you!!!