Add custom -element to wp_nav_menu

The wp_nav_menu() function has an items_wrap parameter which contains this by default:

<ul id="%1$s" class="%2$s">%3$s</ul>

So you can pass it the above default plus any appended string:

$search_form = get_search_form( false ); // Return not echo

$items_wrap = '<ul id="%1$s" class="%2$s">%3$s';
$items_wrap .= sprintf( '<li id="searchItem">%1$s</li></ul>', $search_form );

wp_nav_menu( array(
    'items_wrap' => $items_wrap,
) );