@Jwlewis82 - The Rachio looks wired up correctly. The 24 VAC for the Rachio is the black plug in the lower right hand corner of the last picture from the Rachio power supply.
When was the last time it rained at your location? The reason I ask is that some installers will put the rain sensor in-line with the common wire, instead of using the sensor port. If this is the case, the common wire will be open and there won’t be a complete circuit.
I’m assuming the four LED lights on the Rachio are all lit? Have you tried using the manual buttons on the face of the Rachio to activate one zone?
Finally, just checking that the water is actually on to the sprinkler system. Can you manually activate a valve in the valve box?