This IS the "other hardware" party of the forum, though usually we prefer topics to be somewhat Arduino-related...
You could in fact replace the entire circuit up to the ULN chips with an Arduino or AVR chip, simultaneously gaining the ability to operate via USB (and/or get debugging info via USB.)
it doesn't run when we provide the voltage
"doesn't run" how? What are you doing to it, and how are you expecting it to behave?
You have no bypass caps.
The 74LS05 has open-collector outputs and really isn't well-suited to use as a general purpose inverter. Though it might work if you're using "real" 74ls series logic, since inputs float high.
It looks like the circuit should toggle the relays when you provide a clock by select/deselect/etc the inputs to the '154, but your circuit isn't providing any toggling. Doing so by attach/detach of wires will have "bounce" issues.