To setup what you had mentioned, you will likely need to learn a little bit.
You can create a secure admin panel, or otherwise upload/remove the .php script which will use a 'while' function to mass send your mail to the mailing list.

You will however have to collect those mails into a database first..
Including something like the following into the mentioned contact form after setting up a database will let you acquire those.

<?php
include("config.php");

if(isset($_POST['subscribe'])) {
$submail = $_POST['email'];

$result = mysql_query("SELECT email FROM subscribers WHERE email='$submail'") or die(mysql_error());
if (mysql_num_rows($result)==1) echo " ";
else {
$sql = "INSERT INTO subscribers (email) VALUES ('$submail')";
$query = mysql_query($sql);}
}

?>