Can you finance a car and still make payments on it even though you don't have insurance on it?

Yes one can, but almost all companies providing financing will require you to obtain and have insurance with specified levels of coverage immediately after purchase (if not before even approving the loan) if you don't already have it to protect their investment. If you do not get the insurance required within their time limit, they have the right to call the loan and demand its full repayment or repossess the car.

For example the CU I financed my first car with wanted proof that I had made at least an initial payment on insurance with $250 collision and $100 comprehensive coverage before even approving the loan.

So while it can be done its rarely practical to do. Its really up to the lender.

Of course, your state will still require either liability insurance or posting a bond for the car to be legally driven! But thats a separate issue from financing when you don't yet have insurance.