I've bought multiple cars out of state, and financed through my credit union. Some of the details will depend on what type of temporary or travel permit your state offers; my state (Arizona) has temporary permits that can be printed from your home computer. I've taken the permit to the state where the car resides, paid with a credit union check, and driven home. Once at home, I take the car to DMV for a VIN check and registration. One benefit is that I didn't have to pay sales tax in the state of purchase, nor does Arizona require it.

Another option is to have the car shipped to you, which I've also done. FexEx the check to the dealer holding the car, coordinate with the trucking company, and have it delivered to your door. In my case, the savings on sales tax more than paid for the shipping.

Again, some of the details will depend on your state's rules on temporary registration, but I can't imagine why there would be a problem. And since the car will be registered in your home state, the credit union should be fine with it.