You must make sure you are connected to the GT servers before you can gift a car.
When you have been playing Gt Life you will disconnect from the servers. You can check this by clicking on the community tab and try to look at the log of one of your friends. If it does not show your friends there then you are not connected.
Go back to the main menu and re-enter GTlife and you should re-connect to the servers. Then you should be able to gift the car.
Sometimes you may gift a car and it still not get there. Happened to me yesterday and I lost my Minolta. It said operation complete and the car dissappeared but it did not arrive where it was sent.