Looking at the pictures there, it seems everything is running on 240V. That said, it looks like the relay is a larger one, suitable for controlling a pump. If you’ve got smaller valves for the actual sprinklers, then you should see what those run on.
The open sprinkler, depending on which model you get, can operate most low voltage valves that need anything up to 24VAC. Looking at your diagram, the existing relay uses a 240V coil voltage (this is what operates the relay), so this would need to be replaced with one that has a 24VAC (most common OS voltage, or whatever you end up choosing) coil voltage.
Worst case, you’ll need to replace the valves controlling individual zones, if there are any. You’ve already got the wires run, so that’s the biggest job out of your way.