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You need to contact the credit union and find out what their policies are.
There will be all kinds of terms and conditions the dealer needs to know, maximum loan amount, required down payment, LTV, make/model/year/mileage restrictions, etc.
A car dealership in my town told me they work exclusively with my CU. It could be the case with yours. If so, the dealership should be able to pull up the paperwork online. Call your dealership and ask.
Are you working with NCSECU? If so, I'm well-versed in the process. What you need to do is find the vehicle you like, check NADA and make sure the value isclose to what they're selling the car for (we only finance up to 110%), get a bill of sale, and head back to the Credit Union branch. At that point, we will finalize the loan, you add the car to your insurance and fax us the binder with us as lienholder, and you're good to go back to the dealer with your check to pick up your keys
If you prefer working with the contact center, give them a call and they can handle the entire process over the phone by sending the dealer a letter of gaurantee, then fedex'ing the check that day.