Yes, Coda has a public API available to all users. The API allows you to get data in and out of tables in Coda as well as access computed values from formulas and user-selected values from controls.
The Coda REST API allows users to programmatically interact with Coda docs in a number of ways including:
List and search Coda docs
Create new Coda docs
Discover pages, tables, formulas, and controls
Read, Insert, Upsert, Update, and Delete rows
Access formula values
Update page metadata
Check out the API documentation here.
To see examples of the Coda API in action, take a look at this Coda doc.