If the site adds some markup, Google could send users to the site's internal search page. Apparently, Amazon implemented this feature or Google used Amazon's site to test it. This page explains how to add the schema.org markup: "This markup should reference the URL pattern for your site's search queries in the target property of SearchAction. The URL pattern must point to the same domain that hosts the content being searched. It must also include a variable name in braces that is a placeholder for the user's search query."
Not all the sites have their own search engine, so Google suggests to create one using Google Custom Search. It will still use Google Search, but you can customize the search engine, change ranking, search multiple sites and even monetize it.
I don't think it's a good idea to send users to the site's internal search page, at least not as a default option. Bing already does this and it's probably useful for sites like Amazon, eBay or YouTube, which provide powerful tools for refining results. For most sites, Google's search feature is better. Now the search box will no longer provide consistent functionality: for some sites, Google will send users to their own search pages, while for other sites, Google will show its own results.
Actually, there's a way to tell if Google sends you to a different site when clicking the search box: the message inside the search box is "search site.com" instead of "results from site.com" and the list of suggestions includes an option to show "results for [query]", which sends you to a Google search results page.