Google is a global search engine, and the shop sends the vendor data to google to get the stock status – it does not appear from nowehere. There is an API on the google search console, where the stock data can be fetched from either the OpenGraph protocol or via other API.
The stock information is not available on your local shop, so there is no way to access that. Ajax search pro would have to crawl the destination URLs to fetch Open Graph information – which is not what it’s designed for, it is a local search engine for the local database, and the stock status is not available, even Woocommerce can’t tell if the product is in stock.
Google has the ability to crawl external sites – even if we implemented such feature, that would very quickly cause huge problems. Crawling external sites requires access permissions, very powerful servers, AI interface for data organization etc.. Your website would start slowing down at first, then the database would start growing uncontrollably, not to mention the external site firewalls – your server would very likely end up on a blacklist too.