No credit card required
Browse credit cards from a variety of issuers to see if there's a better card for you.
Your score is the determining factor for your interest rate, so inquiries do matter, because they have an effect on your score. But, it's not a major factor.
When it comes to auto loans, there are companies that will shop interest rates and offers for you. They will pull your credit and give you a list of CUs and banks they deal with and their interest rates they are offering you based on your score. Blue Harbor comes to mind, they are peddled by Credit Karma.
To obtain the BEST interest rate and avoid too many new inquiries, I would secure financing before you go to the dealership. I agree with the others that a CU, over a large bank, may be your best choice for the lowest interest rate. It never hurts to garden before making a large purchase and allowing some of your inquiries to fall off. Best of luck OP!
I have had a different experience with Chase for my auto loan. It may because they are the finance company for Subaru and they gave me a competitive rate beating my pre-approved USAA financing. Chase may be different if apping for a car loan not affiliated with the maufacturer like Subaru.
Right now Chase is from 3.65% to 4.15% in my area depending on if you bank with them or not. PenFed is 1.99% for the same 60 month loan.
Not too mention Chase auto finance customer service is awful. Been there done that.