Here is a picture of the back of the board.
This product does not surface the C wire connection. Nor does the latest model (did those people not get the memo???)
You would have to unhook everything (good luck) and hack it off the back of the board, as Greg Hill describes.
Avoid all the terminals on the lower part of the board. They all contain 120V power. The W and R terminals there are the wrong ones and will set your thermostat on fire if you touch them.
Here is the schematic of the latest board, noting the various wires. You will also need to figure out which one among the identically marked T1, T2 and T3 pairs is actually R. Because if you confuse R and W, hooking up C won't help.
You'd have to inspect the green PCB and add a terminal or solder on a wire to get a C connection on this board -- it doesn't have a terminal ready for you to use.
Pull it from the housing and look at the copper traces going between the thermostat terminal block and the transformer. From one of the T1, the T2, and the T3 terminals there'll be a trace going back to one of the pins below the transformer. That's the R terminal. The other low-voltage pin on the transformer is the C terminal. Its trace will probably lead to the coils of the three relays.
Take care that you don't choose one of the line voltage terminals on the transformer! Follow the traces from transformer pins back to the line voltage terminals at the bottom -- those transformer pins aren't the one you want.
Edit: with the photo of the rear of the board now included in the question, the circuit can easily be traced. I've marked the area on the rear of the board where mains voltage exists in orange -- keep your wiring clear of these. The R and C circuits are also marked.
The C circuit passes to the front of the board via the leg of a resistor. I suggest that you solder your wire on the top side partially so it's easily visible when somebody looks at this board again, and partially so it's easy to verify that it is keeping clear of the uninsulated mains solder points all over the back side. Also note that the white wires are currently connected to the R terminals. It might make sense to swap the reds and whites.
Passing your new C wire through a pair of holes will help relieve strain on the solder joint. Take care not to hit any of the traces while drilling the holes. It's probably easiest to choose the location if you drill from the back side of the board above the trace marked C.