Could you show us a link to your site? Also could you be a little more specific about the ‘zip code search’? With some editing you could add a normal search but that would search your whole site for anything it finds.
If you want anything more specific than that I’m afraid more advanced programming would be required.
You can easily add the form HTML via a widget, but you need to be comfortable with CSS to correctly place and format that HTML in the header area.