Contact form within a Boostrap pop-up modal window - php

Question

I am using bootstrap to open up a modal form.
I really cannot figure out what is going on. This code works in other projects (http://phpform.webitwebsites.co.nz/newtest/) and sends the mail no problem. But if I copy it into my website, it is like my submit button is unresponsive and it doesn't load the initiate quote.php file.
Here is a live version where it is not working
http://phpform.webitwebsites.co.nz/test/
My HTML:
<button href="#freeQuote" role="button" data-toggle="modal" type="button" class="btn btn-primary btn-lg outline" style="font-size:16px;">Free Quote</button>
</center>
<div class="modal fade" id="freeQuote">
<div class="modal-dialog">
<div class="modal-content">
<div class="modal-header">
<button class="close" data-dismiss="modal">×</button>
<h3 class="modal-title">FREE Quote</h3>
</div>
<!-- close modal-header -->
<div class="modal-body">
<h4>Send us a description </h4>
<hr>
<form name="contact" method="post" action="quote.php">
<label for="name">Your Name</label>
<br>
<input class="col-md-10" type="text" name="name" class="input-xlarge">
<br>
<br>
<label for="company">Your Company</label>
<br>
<input class="col-md-10" type="text" name="company" class="input-xlarge">
<br>
<br>
<label for="email">Your E-mail</label>
<br>
<input class="col-md-10" type="email" name="email" class="input-xlarge">
<br>
<br>
<label for="message">Details</label>
<br>
<textarea class="col-md-10" name="message" class="input-xlarge"></textarea>
<br>
<input class="btn-sm btn-success" type="submit" value="Send!" id="submit">
</form>
</div>
<!-- close modal-body -->
My PHP
<?php
$myemail = 'example#gmail.com';
if (isset($_POST['name'])) {
$name = strip_tags($_POST['name']);
$company = strip_tags ($_POST['company']);
$email = strip_tags($_POST['email']);
$message = strip_tags($_POST['message']);
echo "<span class=\"alert alert-success\" >Your message has been received. Thanks! Here is what you submitted:</span><br><br>";
echo "<stong>Name:</strong> ".$name."<br>";
echo "<strong>Company:</strong> " .$company . "<br>";
echo "<stong>Email:</strong> ".$email."<br>";
echo "<stong>Message:</strong> ".$message."<br>";
//generate email and send!
$to = $myemail;
$email_subject = "New Quote form submission";
$email_body = "You have received a new message. ".
" Here are the details:\n Name: $name \n ".
"Email: $email\n Message \n $message";
$headers = "From: $myemail\n";
$headers .= "Reply-To: $email";
mail($to,$email_subject,$email_body,$headers);
}
?>