Hi,
They would only show a document icon if the search is returning a “Post” post type.
As you are using a plugin SearchWP to search for different posts types, the search results now return attachments(media post type) which is what the PDF’s are saved as.
You would need to hire a developer to integrate the custom search result layout for media type post types such as PDFs as unfortunate this customisation is beyond the scope of support.
You could simply insert your PDF links into your Posts, these would then display as expected in the results.
Thanks.