You do not need a stepper motor controller to control a stepper motor.
Probably you do not need even a computer for stepper motor control... but it makes this task a lot easier.
So, it is NOT impossible to control a stepper motor with a device which is able to send a pulse train of programmable frequency, but it is much more comfortable to use a stepper motor controller.
Basically, usually a stepper motor controller off-loads the task of programming the pulse train, including variations of frequency for accelerating and slowing down the drive. It usually provides a software interface so that you do need to calculate exact values of pulses and frequencies when moving the drive to a certain position. You can do all this "manually", it depends on the time and money you can spend.
Also, there is a difference between a stepper motor controller and a stepper motor driver. A stepper motor controller generates all the signals to control a stepper motor and can receive feedback signals such as limit switches, encoders etc. A stepper motor driver "converts" the control signal it receives from the stepper motor controller into power signals powering the coils of the stepper motor. However, powering the coils in most cases is not just a matter of switching on and off the power to the coils but also requires regulating the current through the coils (this is absolutely necessary when using micro-stepping, i.e. positioning the motor to positions between the "angular grid" determined by the number of poles and/or providing more smooth operation when turning).
You can do simple full-step operation by using solid state relays driven by digital output lines, although I would rather recommend an integrated so-called H-bridge driver (which can also be built with discrete transistors).
