The original poster had two questions, but confused them.
1 - "Does anyone have the LIN stack..."
They're asking if NXP provides a LIN App Note with sample code to run the SCI so they can get it working quickly. Forcing engineers to write code directly from the register definitions in the User Manual is so 1980's.
I would suggest searching NXP's site for "LIN Application Notes" and see if any app notes and sample code has been written for OTHER CPUs (including the ARM, MPC and Kitetis ones). The "TRK-KEA8 StarterTRAK" might come with sample code.
2 - "will I need to use something like an ST L9637 to handle the communication?"
Secondly, like most other off-board hardware interfaces (Ethernet, CAN, RS-232, RS-432, USB) an external transceiver is needed to convert the delicate 3.3V signals from the CPU into the ones required externally. In the case of LIN, 0-12V with lots of protection against nasty levels and spikes. We use the ATA6663, but the L9637 would work the same. NXP even has LIN Transceivers like those shown here:
https://www.nxp.com/products/interfaces/automotive-lin-solutions:MC_53488
The above pages should reference kits where you might be able to find sample code.
Maybe NXP Tech Support could provide links to other pages like this?
Tom