It seems it's not using ChatGPT but rather the "standard" GPT-3 models, which do have an API. I didn't look at the code, but they explicitly mention the need for an OpenAI API key.
I've been using similar workarounds to use ChatGPT programmatically and it's just not worth it. In my experience you just get blocked after a couple of requests. I'm really hoping they release an API soon.