This searchbox is powered by Google and has been set up to search within the Coffeecup.com website. You may want it to do a search somewhere else, so you need to go to https://cse.google.com/cse/ to create your own searchbox. You will find easy instructions on the page. You can have it search a site, or just part of a site.
When you have typed your URL and clicked on the CAPTCHA, you get to the page where you can get the code. There you can either grab the whole code and make your own searchbox, or look for the 'Search Engine ID', and copy/paste it into the existing script that you find in the Element pane, HTML window, overwriting the existing ID, which is '003692994821310631992:vma0gnpmvec' (without the quotes)