I have a number of folks at companies, organizations, institutions, and government agencies come to me saying that they want to do APIs and they need some help. In many of these discussions, the first task centers around addressing the motivations behind this declaration and helping folks realize they are already doing APIs; they're just not doing it in any sort of coherent and organized fashion.
Any business, organization, institution, or government agency is moving machine-readable data between internal and external systems in 2016 for use in a variety of applications. This is done in a variety of often proprietary, ad-hoc, data dump, etc. approaches that are often dictated without any coherent vision. API does not always mean 100% REST. It is about establishing interfaces between systems for programmatic usage across a variety of applications. The trick is to do all of this in a simplified, standardized, consistent, and low-cost way modeled on what has been working for the web.
You are already doing API. You need to identify all the different ways you are already exporting, importing, migrating, syncing, and putting data to work across operations and begin to establish a coherent roadmap and approach to communicating about all of this. Then, get to work identifying where the opportunities for standardizing how everything is defined, how access is needed, and a baseline for communication around these critical integrations on an ongoing basis. It's not about whether you should do APIs or not. You are already there. It's about how to start doing it in a more organized and coherent way.