Dear jonathanmccarron,
In the SDK from the LPC55s6x there's an example of how to use the IAP function, if you check the fsl_iap.c you will see the next define:
#define BOOTLOADER_API_TREE_POINTER ((bootloader_tree_t *)ROM_API_TREE)
This define will provide you a user application access to APIs exported by the bootloader.
If you check the bootloader_tree_t type, you will see a function called runBootloader.
I hope this helps you.
Best Regards,
Alexis Andalon