This is what the SearchWP person told me:
“In order to implement something like that you would need to install SearchWP on each respective website, and then depending on whether the visitor chose to search the Catalog or Website you would redirect to that standalone website and pre-populate the search parameter. That’s what’s happening on the site you linked when you choose to search the Catalog; note that when you search the Catalog a new browser window/tab is opened to a URL that pre-populates the search you entered, whereas if you choose to search the website, you are brought to the standard search for that site.

So yes it is technically possible but it’s really only a small bit of form wizardry to make it happen as opposed to SearchWP doing it on its own.

SearchWP isn’t able to index content from an external site, however, so the implementation would need to be the same as the site you sent where one choice opens a new tab to a different site, and the other choice sends you to the standard search results.”