Yes a car can and will start without the alternator being installed or without the alternator being hooked up.
But once the car gets started without the alternator the car will use the batteries electricity to keep the engine running for the spark.
When the engine is running without the alternator the battery will eventually drain and the engine will die out.
The engine needs electrical current to keep it running because the ignition coils need electricity from the alternator and battery to keep the engine started.
So you could drive a short distance with a bad alternator but eventually the battery would drain and your engine will die out.
I have started vehicles up without the alternator before and they do run for around 30 minutes or so until the battery drains.
The alternator not only recharges your vehicles battery but also keeps the vehicle running.