Creating a custom multilingual form

What would be the easiest, cheapest approach for this?

Simple, create one form per language.

In functions.php or, preferably, as a custom plugin:

add_shortcode( 'my-lingo-form', 'shortcode_wpse_98360');
function shortcode_wpse_98360()
{
    $lingo = your_language_detection_method();
    switch( $lingo )
    {
        case 'en':
            echo do_shortcode('[form-en]');
        break;
        default:
            echo do_shortcode('[form-other-languages]');
        break;
    }
}

In your post or page: [my-lingo-form].