Comment on WordPress Header Images by SEO Dave.
You can probably add some code to the file that holds the search form code.
The CSS for the code in that file is in the main style.css file:
If you wanted to remove the search form code (if you want the search facility add it as a widget on a menu) and replace it with an email link etc… you’d delete the form part of the searchform.php code (so keep the div bits so the positioning doesn’t break).
You could either add inline css to your email link or add it to the styling above (if it needs any styling).
You could also add your email links above or below the form code in the searchform.php file to have the email link above or below the search form.