Is there any way to include documents I created with the media core module into the search form of the page? I only found the categories "users" and "nodes".
As I understand, you want to search by Media entities/records.
If you use the standard Search module from Core, then it impossible.
But you can install the Search API module, configure the search server (search server can use current drupal Database or Apache Solr, or smth other).
In the Search API you can create indexes and include the documents into the index.
To display a page with search results you can use a Views module (using content from Search API index) or you can try to use the Search API Pages module