Any API would have to be developed 'in-house' to keep the code private (or at least under NDA). The purpose of an API of this sort (read-only) is to expose endpoints which external devs can access to view data, ideally in a nice digestible format like JSON. Said devs can then display said data in a nice pretty format on a web page or in an app, for example.
The data can also be used for analysis. Things like examining which enhancements are most popular on endgame Mind Controllers, for example.
What I did with Tyto, my DCUO app, was provide users with the most efficient gear purchases for their current combat rating, based on cost, highest CR boost, etc. A similar algorithm is plausible in CoX, based on all the data you could fetch across the system.
But I digress.
A periodically updated shadow database, where only the changes since the last update are pushed, served separately from the game servers (e.g., every 15 minutes, the game servers would update the shadow) may mitigate lag concerns. I don't think anyone would expect second-by-second updates, nor would it be reasonable to do so.
The API would then allow read-only, rate limited access to that shadow data.
It's all pretty standard stuff, really. It just doesn't currently exist, for reasons over which we can only speculate!