This sort of stuff is pretty common. I have a 6 station unit but it also has the common supply for a pump as you require, and it runs from a 24 volt external transformer. I use my system for watering the vineyard.
Each of the outputs is switched with a small TO92 case triac, and the current capability is 9VA max. At 24 volt this equates to a current of 0.375 Amp. From memory, my solenoid valves are about 4 or 5 VA each, so you shouldnt have any problems with driving a relay from each output.
The 'Watermaster' thing I have is has a model number of '57116'.
Its probably from the same factory as yours!!