Hi Aaron I think this thread on stackoverflow can be of some help to you basically here they are attempting to write a backfill script that will pull in historical follower numbers for linkedin companies. You can also start conversation there.
https://stackoverflow.com/questions/27973204/linkedin-historical-company-statistics-giving-me-the-wrong-number-of-followers
Recently i was also working on the a related aspect where we required to pull in data from linkedin Api for local services provider small companies on linkedin for my website, working on it currently,
Happy to help you let me know if you have any questions.
Thanks
CMO - UMC (Toronto)