Duplicate product search box from one site and add to second site

We use a third party site to run a parts shopping catalog. The site uses a search function that auto fills inventory as you type. I would like to add the search box from the third party site to our main site so visitors can search the catalog from the main site. So write code that will query the search function on the third party site bases upon search terms entered on the main site, then return the results from the third party site to the search form on the main site.