It is not possible to integrate with the core WordPress search because it is highly optimized for only searching WP pages/posts or Custom Post Types (CPT) if they are registered for search.
Connections does not use CPT for its data because the db schema for CPT’s are optimized for post like data only. Connections uses a db schema optimized for directory content for performance and scalability. You’ll find that much larger plugins such as WooCommerce and Easy Digital Downloads have moved or are moving away from using CPT’s because using them are too inefficient for the custom data schema’s they require.
If you want search, you need one of the premium templates. If you want to have a search input on your home page, use the Search Widget which comes as part of the Widget Pack. You can place this widget in your theme’s sidebar so it shows up everywhere.
Hope that helps!