Q: What's the best way to consolidate $31k credit cards no equity available but have a $20k car unfinaned?
A: I would suggest 0% interest balance transfer to another card. This will generally give you 1-2 years without further interest accruing.
Transfer the balance, cut up all cards so you can't keep spending, pay the debt off before the zero interest period ends.
Q: Can I purchase a car through my business and claim the repayments?
A: You may be able to but should check the details with your accountant.
Depending on the product you may be able to claim some/all of the interest expense and depreciation on the car, or the whole monthly payment as an expense. There are plenty of options that may suit what you're after.
Have a look on at the CBA asset Finance page if you're after more information.