Forgive me if this doesn't make sense:

If I'm implementing search in an application and want to use NLP, do I need to train the search or are these solutions already ready to go? I'm not sure how other people do it/how search works/if you need to tell it what to do.

Well, of course the engine needs to have access to some corpus to search on. So the general answer to your question is: yes, however this step typically not called "training" but "indexing".

Most engines will repeatedly index your contents with crawlers or similar.