The goal is to link to a page displaying all the products that come up when you search the term "tangy".
As you can see on the first website i tried creating a form to pass the search parameters to the second website but without success...the "tangy" products do not show up when you complete my form and the page is not displaying correctly...
Contact the company directly.
You cannot blindly pre-populate form fields unless the website happens to support it. maybe it does, maybe it doesn't. You can try the obvious (URL parameter or Form Field that matches the ID or NAME of the search field)... that might get it to pre-populate, but even that won't necessarily trigger the search.
Faster and more reliable solution - contact the company with what you are asking to do. Most sites will be happy to help other sites that want to drive traffic to them. If they don't already support the feature you are requesting, it may be something small enough for them to do that it may be worthwhile for them to make the change based on an estimate of the volume you expect to send them.