Yes I know why you are not allowed to place link to site but I am also not suggesting the Google Search but there are 2 ways a form is submitted

GET and POST

If they support Get you can use the direct URLs

Else if they don't then you would need to get form and use the same code for the button click and get a response.

This means you create a page on your site which can be made to click and then when clicked it gets a results from that site into some frame and then do accordingly.