Can a car dealership sell your trade-in if your new car financing doesn’t go through?

There is no single answer to this question: it depends upon how the purchase of the new car was set up. If the sale agreement made the sale contingent on financing--similar to how a home sale contract could have a "finance contingency" making the consummation of the sale dependent on obtaining a mortgage--then if the contingency is not fulfilled--that is, the financing is not obtaining--then transaction will be voided. In that case, each party gets back what it had contributed or received from the deal--e.g. the buyer gets his trade-in car back, the dealership gets the new car.

However, unless the financing was an actual contigency of the sale, the sale will not be voided by a failure of financing. In this case, the dealership may sell the trade-in, apply the proceeds to the sale price and, if necessary, sue the buyer for the balance. This would be similar to a home purchase which is not contingent on mortgage approval; the buyer needs to then find the money somewhere.

So the answer to what happens in any specific case will depend upon how the deal was structured, which answer should be found in the contract of sale and in any representations made by the dealership prior to the purchase, which representations were material (or key) to the buyer deciding to buy the car.