Is there a way to run the model locally? I'd rather not have to pay a monthly fee, if possible.

You don't pay monthly fee to use OpenAI models through the API. You pay per token. https://openai.com/api/pricing/

There are other CLI coding agents like Aider (https://aider.chat/) that will let you point at any model. The problem is that local models are dramatically worse than these big hosted models — I have not seen anyone claim they are good enough for these kind of tools. See https://aider.chat/docs/leaderboards/

Why would they give you the model?