As of June 2019, FPbase is moving to a GraphQL-based API. GraphQL is a modern query language that gives clients the power to ask for exactly what (fields) they need and nothing more, and makes it easier to evolve APIs over time.
FPbase has a REST API for retrieving data in JSON or CSV format. It is very basic and still in development (unstable). Until we know there are people who would like to use it, there are a lot of other things that also need attention. So let us know if you'd like to use it!
NOTE: FPbase is moving away from the REST API, and all development will use the GraphQL API mentioned above.
These docs are incomplete and potentially wrong... In the meantime, there is (slightly) better information at this page: