Can a car dealership reverse a car sale if the lending company removes their approval after the sale?
If the lender decides not to send the agreed funds then there is no sale to reverse.
X signed a sale contract with the dealer in which they agreed to pay a certain amount of money in exchange for the car. X elected to make payment with funds obtained from a loan, through a separate, but related, deal between herself and a lender. X, the lender, and the dealer also all agreed that the lender would transfer the loan amount directly to the dealer, instead of it going through X as a middleman. That is convenient for X, but it mostly protects the dealer and lender.
The dealer also had sufficient confidence in X's funding that they allowed X to take physical possession of the car before the funds were actually delivered. That's routine. But I'm sure you would find that the car's title had not yet been transferred to X. Until the dealer transfers title, they own the car, not X. X may even have explicitly agreed, among all those things they signed at closing, to return the car if the funds were not delivered.
If the lender refuses to provide the funds after all then they may or may not be within their rights under the loan contract, but either way, X has failed to fulfill his obligations to the dealer under the sale contract, so there is no sale. X does not own the car in any sense, and preventing the lawful owner from reclaiming possession would be theft.
I would expect the deal unraveling also to mean that X gets back any down payment she paid. (This will have been paid to the dealer, not the lender.) If X is owed that, then I think it would be reasonable to insist on receiving the refund at the time he returns the car. However, it is conceivable that among the things X agreed at the closing table was that the down payment was not refundable. She should read all the paperwork carefully to see where she stands.
The deal is finalized. All the contracts are signed. The customer drives the car off the lot.
It's more precise to say that the terms of the deal are finalized, as evidenced by the contracts being signed. The deal itself (at least two separate deals, actually) is not complete until all parties have fulfilled their obligations.
The salespeople even say “Once this is finalized, we no longer have anything to do with the car; paying back the loan is directly between you and the lending company”.
This "finalized" is not the same as the previous "finalized", though the dealer may well intentionally obscure that. This "finalized" is more or less the lender fulfilling their agreement with X to pay out the loan proceeds to the dealer, which typically has not happened yet when X drives off the lot. When the lender does pay, yes, the dealer is then in the clear as far as the financing. Once they transfer title, the contract between them and X is fulfilled, and the dealer is out of the picture.
They also mentioned that they don’t have a grace period where you can return the car, the sale is final.
That may or may not be true, but it doesn't speak to the question. That the contract and applicable law do not give X the option to back out of the sale would not speak to the dealer's options under the same contract and law.
For what it's worth, I think most U.S. states do provide a window in which buyers can back out of retail new car purchases, but the same is not necessarily true of used car sales. I don't know about Utah in particular.
Can the dealership reverse the sale of the vehicle like this?
They are almost certainly within their rights under the sale contract to demand return of the vehicle on the basis of X having failed to fulfill the terms of sale, but one would need to check the contract and all secondary agreements between the two to be sure.
If the lender “reverses their approval”, doesn’t that mean they technically paid for the purchase of the vehicle, but then later told the dealership, “We changed our minds” - so isn’t this an issue between the dealership and the lender?
No. The lender did not agree to pay for the car on X's behalf. They agreed to extend X a (secured) loan. That's a deal between the lender and X. There are contingencies, such as X spending the money to buy the car that is to secure the loan, but that does not change the fact that it's a deal between them and X. When the dealer is paid, it is with cash, not with a promise. If the lender doesn't provide that cash then no, there is no sense in which they paid for the car. There is then an issue between the lender and X, but the primary issue is between the dealer, whose car X is holding, and X. There is no particular issue between the lender and the dealer, though the dealer is probably unhappy that the lender scuttled the deal.
Isn’t the vehicle now the legal property of the customer?
Has it been paid for? No.
Has the title been transferred? No.
Is it X's legal property? No.