I'm not very familiar with Exploration Buddy. It's possible that it does not have an API?
It does read from your local journal files and lookup information from EDSM.
EDSM has it's own API. You can check out https://www.edsm.net/en/api-system-v1. There are additional API endpoints available from a menu in the bottom right corner.
Yeah, normally you key the API key from the website in question and enter that into the program. When you join the site, like EDSM, it will generate an API key, that is then entered into the program where it asks for the API key for that site.