Can a car company refuse to give me a copy of my payment history and balance details?
The car company (or anyone else) is not responsible for acting as your record keeper or bookkeeper - that's your responsibility. So, no.
That said, I cannot think of any reason whey they would refuse such a request. They may be able to request a fee for doing so.
Of course, if you were in dispute with them these are certainly things you could subpoena.