A thought,

The problem seems to come on when there is AC on the contacts , going through the sprinklers.

This says to me, that its a mains spike getting onto the micro controller,

I assume there is no direct connection between the micro controller and the AC / sprinklers,
could it be that the sprinklers and the micro are on the same mains socket ?
and its a mains spike that's getting into the micro via a low quality power supply for the micro ?

If so, try plugging the sprinklers into a socket on one side of the house, the micro into another circuit, / other side of house

if it fixes problem, then you need a better PSU for the micro.

 

If those sprinkler valves operate with 24 volts AC and if that relay board actually has adequately rated relays then usually snubbers are not required in most applications. But since it appears to be a sparking at the relay contacts issue here, a snubber rated for 24 volt service should be cheaper. Probably 0.047 mFd and 100 ohms. Or whatever is available.

 

I took a look at this thread:

Thoughts.

#1. get the ground and (+5) out of the ribbon cable. Run those wires separately.
#2. The RC snubber across relay contacts are desired for AC arc suppression. They are available as a single component as well.
#3. Watch cable routing.
#4. Remember that shielding reduces RFI and twisting reduces EMI. You might have RFI.
Twisting and differential signalling is best. I doubt you have that problem.
#5. Put a ferrite bead on the power to your web server and relay power board.
#6. try a different 5V power supply temporarily.
#7. Turn the 24VAC power on with a zero-crossing triac switch.