How do I wrap 3rd party jquery scripts so that they work in wordpress?

As I stated in the comments. You need to have jQuery and jQuery UI loaded in order to use the autocomplete UI. I modified your hook to have dependencies on the required libraries. Now all you need to do is set the path to your custom js file that initializes autocomplete. The dependencies will load the files needed as this is built into WordPress. You can see the available libraries here: wp_enqueue_script

function add_qa_scripts(){
    wp_enqueue_script(
        'my-autocomplete',
        PATH_TO_CUSTOM_JS_FILE,
        array(
            'jquery',
            'jquery-ui-core',
            'jquery-ui-autocomplete'
        ),
        '1.0.0', // version of custom js file
        true // always load in footer unless absolutely necessary
    );
}
add_action('wp_enqueue_scripts', 'add_qa_scripts');