Total cost under $20 would make it very difficult to do a sun tracker.
However if you use a cheap RC servo for the mechanism, (e.g.
http://www.ebay.com/itm/9G-SG90-Micro-Servo-motor-RC-Robot-Helicopter-Airplane-Control-Car-Boat/272362301472)
it may be possible, if you can get the photocells cheap enough.
The varying width servo pulse could be generated by an OPAMP circuit or by a CMOS 555 although the latter contains a flipflop so might be too 'digital' for your project supervisor's liking.
The first stage would be a bridge with the two photosensors and reference divider feeding an OPAMP that outputs 0V when the tracking is spot on and a varying positive or negative voltage for the direction and speed of movement required. That feeds into an integrator, possibly a full analog PID controller to generate a control voltage representing the sun's position.
The control voltage (+ a DC offset) is used to generate the servo pulse by altering the charging time of the timing capacitor in an OPAMP RC oscillator, by clamping the feedback. A diode steering network isolates it from the discharge time so that an average period of 20ms +/- 1ms is maintained.
Of course the devil is in the details, but it would be possible to simulate the electronics in LTSPICE to be fairly confident it will work as expected before building it.
The other problem is demoing it indoors or on a cloudy day. You'll need to make sure it can track a torch beam in a room , even under normal room lighting.
Be aware that you absolutely cannot share the circuit here or on any other public forum until it has been marked and your grade assigned, or ask very detailed questions about your specific implementation as any reputable college uses anti-plagiarism software that searches the web for attempts to pass off someone ele's design as your own. If a search for keywords related to your final project proposal pulls up this topic, you'll loose marks under rule 4. If anyone posts a schematic and you 've used it as a substantial part of your circuit, expect a grade of zero!