What is the proper way to cite the usage of an API?
I don't think you'll find any documented correct way to do this. Personally, I would write a sentence saying that I did it, a sentence describing what an API is, and a link to the version-specific API documentation website (i.e., https://developers.google.com/youtube/v3/ for the API Version 3 instead of https://developers.google.com/youtube/ the general Youtube API help website).

If you're including code in the submission, consider adding the API code in an appendix.