Woocommerce product search widget simply lets you add a WordPress widget that can search products you have added to your woocommerce site. I added it to an ecommerce store I created yesterday but failed to work. I deleted it and installed an advanced search plugin for WordPress which is working fine on the store.
So, if Woocommerce Product Search widget is not working for you, delete it and install the "Advanced Search Widget" plugin by Aaron Axelsen. It was last updated 3 years ago but still works perfectly on my ecommerce store.
It simply allows you to search for only a specific post type (eg Wooocommerce products) and also select which fields to query (title, content, tags). In addition, it is also possible to place multiple widgets onto the sidebar to search through other custom post types.
If you want to use it to search woocommerce products, select "product" from the "Post Type" drop down as seen in the screenshot below:
Alternatively, you can try the "WooCommerce Predictive Search" plugin or the Enhanced Product Search for WooCommerce which costs around $49 for a single site.
I hope this helps.