Basically, my preferred way is using some Server-Side language such as PHP/Java and make a HTTP-Request and then parse the page and find the requested select/element.
You've tagged this question as javascript/jquery so I'm guessing you're aiming to this kind of solution.
It's possible. You'll need to make a cross-domain request and parse the results to get the select box value.
Take a look at this question for how to make a cross domain request:
Loading cross domain endpoint with jQuery AJAX
take a look at jQuery API to parse and traverse HTML.
the rest is basically case-specific programming