Getting Drupal 8 to use an external index?
What you could do is use the Migrate module to migrate content from one site to the other. What you need to do is:


Build a custom source plugin for it so that it fetches its data from an external source (i.e. the other site) and map it to a content type in your Drupal install. 
Then set up a cron job to run the migration periodically to update the data on your end. Make sure you set the migration to "update" since by default, migrations won't happen on already-migrated content.
Create your search however you want in your site (Views, Search API, etc.) using that content type.


In D7, this was a job for the Feeds module but they don't appear to have a D8 version yet.