wp_mail doesn’t work when logged in?

You’ve got a typo in the hook that runs for logged in users:

add_action('wp_ajax_search_notify_emaill', 'search_notify_email');

There’s an extra l in the hook name. The hook names need to be the same, apart from nopriv_, so you should have:

add_action('wp_ajax_nopriv_search_notify_email', 'search_notify_email');
add_action('wp_ajax_search_notify_email', 'search_notify_email');

When you send an AJAX request with an action WordPress runs the hook wp_ajax_{$action} (where $action is the parameter passed with the request) if you’re logged in, or wp_ajax_nopriv_{$action} if the user is not logged in. Since you’d used an incorrect hook name for the logged-in version (due to the typo) your search_notify_email() function wasn’t hooked to run when the user was logged in.