The WooCommerce Product Search field can be added via a widget and also via a template tag…
Another option is a filter get_product_search_form() by which you can create your own custom search field including using an icon.
The Original Markup
This will just reproduce the form in it’s original format.
The New Markup
Here a fontawesome magnifying glass is added with the original submit removed
Add some CSS
Floating the icon to the right and a splash of color.
For an ecommerce heavy site, you want the search to use the woocommerce layout and product results only, this works great for that issue.