I do not have a Google API key, so please do not suggest any websites that require one. Essentially, I would just like to be able to monitor weekly, for example, how my rankings are for certain keywords or phrases without having to do it manually. A website or tool that has a chart function would be great especially.

Good question. There is a good ranking-meter at seochat.com. Unfortunately, it is done manually for each of the phrases/words you want to chek out.
maybe, just maybe, they have chart function. good luck in finding it.