Hi,

If you need to search the products with barcode in the website, you have to inherit and modify the function named: _get_search_domain inside the website_sale module.

def _get_search_domain(self, search, category, attrib_values, search_in_description=True):
    domains = [request.website.sale_product_domain()]
    if search:
        for srch in search.split(" "):
            subdomains = [
                [('name', 'ilike', srch)],
                [('product_variant_ids.default_code', 'ilike', srch)]
            ]
            if search_in_description:
                subdomains.append([('barcode', 'ilike', srch)])
                subdomains.append([('description', 'ilike', srch)])
                subdomains.append([('description_sale', 'ilike', srch)])
            domains.append(expression.OR(subdomains))

In the above code sample, you can see that i have added barcode field to search. So you can inherit the above in a custom module and make necessary changes,

For reference:
1. Inheriting controllers in odoo
2. Q Context in Odoo

Thanks