You can use the Wanikani Open Framework, which caches item data, then just fetch the audio URLs from there.
wkof.include('ItemData');
wkof.ready('ItemData').then(get_items);
function get_items() {
config = {...};
wkof.ItemData.get_items(config).then(do_stuff);
}
function do_stuff(items) {
// Do stuff
}