You are going to struggle but I have some advice at hand and I wish you the best.

I'm a 21 year old british student studying in the USA this year and I want to do something similar in the summer. From what I have learned so far, you will be able to buy a car and get the insurance although you need license plates and for which you need to be a US resident. Car rental is possible, even as an 18 year old although you will need to use some tricks to keep it affordable and it isn't easy at all. Will PM you some details.