Will I be able to get an auto loan?

I went to a dealership to buy a new car. I have 765 credit score so the dealership told me id def be able to get a loan thru them with a low interest rate. Well, turns out they told me the scores great but i have limited credit history. They said because i have limited credit history the dealership cant give me a loan thru their bank. They said they should be able to get me a loan thru another bank or credit union. They told me they sent the information to 10 banks and are waiting for a response. I have 3 credit cards with $ 0 balance and a total limit of about $ 20,000. My school loans add up to about $ 20k as well. They said my income wasnt a factor its just the fact that I haven’t had any major loans like a car loan or mortgage before. Clearly I need this to develop a stronger credit history but do you think I should have any problem getting a decent rate without having to find a co-signer seeing as my credit score is very high? What do you think the banks would offer as a rate if anything? (the dealership originally told me theyd get me a loan for 3.9% since i have over a 720 but clearly i cant go by their judgement.)
car is $ 21,000-$ 22,000. 4000-4500 cash down. then after rebates and deal, loan amount is 15,000.

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