Shift Registers Controlling Seven-Segment Display without a Latch?
It can work without a latch providing the unstable period while the data is shifted to all the drivers is small - you probably wouldn't notice the very quick flickering of the displays. Maybe if you could tell us the clock frequency this would help but, if it's running at 1MHz and there are 4 severn-seg displays then that's 28\$\mu s\$ for the update - not a long time at all and maybe the clock is even faster.

Item C on your first drawing holds the answer yet I don't recognize it - probably a GAL device.