I wrote to the FSF, and here is their response:
Given the API key and the source code, people who receive the source code of your software should be able to compile the same binary they got from you and use it in the same way. From your description this seems to be the case.
They should also be free not to use your API key or web service, but to modify the software and connect it to a web service of their own choice or design with their own API keys. I'm unfamiliar with the web service/s in question and therefore I don't know if this is relevant to your specific situation.
I hope this answer is of help. Please feel free to write back if you have any further questions.
-- I am not a lawyer, the above is not legal advice
Regards, Yoni Rabkin