Simple JSON+LD Boilerplate for configuring your Site Name and Google Sitelinks Search Box Schema.

Site-Search-JSON+LD.html

<script type="application/ld+json">
{
  "@context": "http://schema.org",
  "@type": "WebSite",
  // Include your Site name within Google Search Results
  "name" : "James' Donuts",
  "alternateName" : "James' Donuts Ltd",
  "url": "https://www.jamesdonuts.co.uk",

  // Enabe the Google Sitelinks Search Box
  "potentialAction": {
    "@type": "SearchAction",
    
    //Use only one of the following lines
    "target": "https://query.example.com/?s={search_term_string}", //WordPress & WooCommerce
    "target": "https://query.example.com/catalogsearch/result/?q={search_term_string}", //Magento
    
    "query-input": "required name=search_term_string"
  }
}
</script>