The larger banks are seeing a lot of market share in this area, but the larger credit unions and the ones being run by ex-bankers are setting up arrangements where a non-member can finance a car right at the dealership just by signing up to be a member.
The finance manager is going to find someone to approve your loan. It's just a matter of what you're going to pay for it. It may not be the best rate out there, but it will get you in a car.