I've put together a list of websites that I have has success with in the past. This is by no means a comprehensive list, it’s just something to get started with. I would suggest searching on various forums such as Taxi Music, Envato Forum and Music Library Report to learn more about the library music business.
I would also recommend this book The Business of Music Licensing which is a book that got me started when it comes to writing music for libraries.
Writing for libraries can be a tough game, particularly trying to get placements on exclusive libraries. My best advice is to start off with some piece on some of the non-exclusive libraries such as Audiojungle & Pond5. Below is a short list of some places I found good:
In addition to the websites it is also good to understand the business so below is a reading list that I believe can be very helpful.
Hopefully this has been informative, any questions or feedback please feel free to contact me!