Two possible steps:
- "Make the same API request on your web site"
- "Reuse Hovercards MediaWiki modules on your web site"
For 1, Page info in search results mentions how Hovercards does it as an alternative.
For 2, I'm not sure how a third-party site could reuse our CSS and JS ResourceLoader modules. mw.loader.load( 'ext.popups; ) is all I need to make it go on a Wikimedia site...