If an alternator is properly sized for the car and the alternator is working properly then yes the cars engine should still continue to run without the battery installed in the car.
However you do need a battery in the car to get the car started.
But even if your battery is completely dead and will no longer hold a charge once the car is started you should be able to drive the car until you get to someplace to replace the battery or make it back home as long as you don't shut the vehicle off during that time until you get home or to the place to replace your battery.