To store the combination of the first name and last name, just use a simple variable to refer to the full name, e.g.,
// Check that $_POST["first_name"] and $_POST["last_name"] exist and validate them:
// Now construct the full name:
$full_name = $_POST["first_name"] . ' ' . $_POST["last_name"]
Then later you can use:
$mail->FromName = $full_name;
Actually, you don't even need $full_name because you can set $mail->FromName directly, but it may be clearer and you might want to do other things with the user's full name.