If you write a check to the dealer to buy the car, the check is not the gift. You are giving the check to the dealer, not to your daughter. So it doesn't matter what kind of account the check comes from, who writes it, or how much it's for.
The actual gift is the car. You and your husband can't each give her half the car. There's only one car, and it's worth $20,000. Your income tax return is not the issue. Gifts don't go on your income tax return. To stay under the $16,000 annual exclusion you would have to file a gift tax return, Form 709, to split the gift. This is completely separate from your income tax return. You would not have to pay any gift tax unless you have previously made gifts totaling more than $12 million over your lifetime, but you have to file the return. You cannot file a gift tax return with TurboTax.
A "gift letter" would not accomplish anything. The only way to split a gift is to file a gift tax return.
If you want to avoid filing a gift tax return, you and your husband could each give your daughter $10,000, in two separate checks, one from each of you, and let her buy the car. Then you are each making a gift of less than $16,000. It doesn't matter if the two checks come from the same joint account, as long as each of you writes and signs a separate check. The checks have to be payable to your daughter, not to the car dealer.
