That schematic is for a unipolar stepper driver (5 wires) powering two coils at each cycle(Full-step), that gives more power but less speed. To make something to run a stepper driver without a uC, you will need some gates and flip-flops too, and ofc some source of clock, like a multivibrator (555).
I would suggest starting with a 2 bits counter( two T or JK flip-flops can do that, and some ICs have two per package) and then make a truth table with the counter as the input signals, and the four transistor drivers as the output signal, with that, a couple of ANDs and NOTs can make the circuit. What kind of step you will use, is up to you to choose... (wave, full or half-step)