I think the issue here is that page 3 isn't on your server, so I'm assuming a SESSION variable won't help. I'm also assuming you know the POST variables that the 3rd-party page is looking for. If that is the case, then it is just a matter of creating a form full of hidden fields. You can use a form button or Javascript to submit the form, depending on whether you want user interaction.
Here's an example with user interaction:
echo "<form name='form1' action='page3.php' method='post'>\n";
foreach($_POST as $key => $value)
{
// Assuming "submit" is the last form object on page 1...
if($key == "submit") break;
// Or "continue;" if you have other irrelevant fields to skip.
// Do some validation here.
// Create the hidden fields...
echo "<input type='text' name='$key' value='$value'>\n";
}
echo "<input type='submit' value='Submit'>\n";
echo "</form>\n";