A 5v stepper motor can be controlled directly from the board provided it's amperage doesn't exceed the maximum amperage of the Arduino. So yes, it is possible, but you won't get much torque since it would be running directly off the Arduino's power supply. Using a driver of the proper type for the motor you're using will allow the driver to be connected to an adequate power supply and be controlled by the Arduino.
Very bad idea as it is almost guaranteed to exeed the max current supply of the arduino pins.
You can do it with transistors rather than directly from the arduino I/O pins.
By turning on the coils in the right sequence, you can make it spin... Motor driver chips are cheap though and easy to use so not really sure why you'd want to do it without one.