Well, your page B needs to be a php script, if it isn't already. Also, I believe what you want in your input tag is name="email" instead of id, and your form needs an action, like
Code:
<form action="page_b.php"><input type="text" name="email"><input type="submit" value="Continue"></form>
Then in page_b.php (or whatever you want to call it), you would get the value of the email field in $_GET['email'], so you can use it in the value attribute of the input tag. But you would want to use htmlspecialchars() on it to make sure the user didn't enter html into the field, because that would cause an error. So page_b.php might be something like:
Code:
<?php
$email = htmlspecialchars(trim($_GET['email']));
echo '<form><input type="text" name="email" value="' . $email . '"><input type="submit" value="Register">';
I should mention that in the vbulletin code it's done a little differently, so if you are including vbulletin code in your script you might want to use the vbulletin functions.