This plugin looked ideal for searching items in a Woocommerce shop. However, it does not work as expected for the custom post type, i.e. "product" that WooCommerce uses, which is a great shame.
This plugin will fetch the taxonomies of the POST post type and do a search on terms in those taxonomies only. That is hard-coded into the plugin. It does not look at what post type is being searched on, and look at the taxonomies for that (or those) post types.