Can a car dealership ask for a vehicle back if you traded inyour old vehicle, put money down and have a signed finance contract?

More information is needed here.  What is the reason that they gave you for having to return the car?  There has to be a basis for their attempt to void the contract between you here.  Some legal basis such as fraud or misrepresentation or a basis that is written in to the contract terms itself.  Have you breached the agreement in any way?  Not paid the financing?  Then they would be repossessing the vehicle but that would be the finance company and not the dealership (unless they are one in the same).  Either write back with more information or seek legal help in your area on the matter.  Good luck to you. 

The issue isn't just whether or not you have the signed finance contract--the primary issue is whether the payments have been made. If for some reason, for example, the first monthly payment was not made, then so long as the fault was not the dealerships's, they have a right to seek the return of the car, since you are in breach of your contract. It doesn't matter if the check was lost in the mail, the financing division or company made an error, you never sent in something to the financing company you needed to, etc.--if the payment is not made, they may seek the car's return.

On the other hand, if all terms, including payment terms, of the agreement of sale are being met (e.g. they're getting paid what they should) and you did not commit fraud in the application (e.g. you didn't lie about your income), then it is difficult to see on what grounds they could seek the return of the vehicle. Call them first thing in the morning--find out what the problem is.