Is there a plugin that allows a user of your site to get an email from you with an attachment?

you can use wp_mail( $to, $subject, $message, $headers, $attachments ); to send emails with attachments.

So all you need is a simple form

<form name="email-att" id="email-att" method="POST" action="">
Enter your emial: <br />
<input type="text" name="email" id="email"/><br />
<input type="hidden" name="action" value="email-att"/><br />
<input type="submit" name="submit" value="submit" id="submit"/>
</form>

and to process it

<?php
if (isset($_POST['action']) && $_POST['action'] == "email-att"){
   if (is_email($_POST['email'])){
        $to = $_POST['email'];
        $subject = "email subject line";
        $message = "email message body";
        $attachments = array('http://full_url_to_file.zip');
        wp_mail( $to, $subject, $message, '', $attachments );
        echo 'mail sent! check your email';
    }else{
        echo 'Please enter a valid email address ';
    }
}
?>