Unfortunately, I don't have access to a scope. The guy who took the image above was also kind enough to throw together a schematic:
Looking at that schematic you've posted, I would expect to see an electrolytic capacitor after the bridge rectifier and another capacitor at the output of the regulator.
I think they really exist, looking at the photo, but aren't shown on the schematic.
There is probably another resistor/capacitor in the thermistor circuit where it connects to the microcontroller, let's just assume (as discussed in your other posts) that this is somewhere inside the "microcontroller block" as shown on your schematic.
What voltage do you get after the rectifier/capacitor, at the voltage regulator input? I would guess something like 12V. Or just look at the relay, which should probably answer that question. What's the relay coil voltage rating? If you get the relay model, then the current requirement can tell us whether the transistor is appropriate or what transistor/resistor you should use to drive it off the Arduino (or similar).
And what's the voltage at the voltage regulator output? That's 5V, right? If that's a 7805, or equivalent, using it to drive the Arduino's 5V rail should be fine.
I suppose it's possible (but unlikely) that one of the SOT23 components on the microcontroller board is a 3.3V regulator and the microcontroller is outputting 3.3V logic levels. I doubt it, though.
OK, so you've got your relay coil connected to the 12V (or whatever) unregulated rail, and connected to the usual sort of NPN low-side open collector switching transistor, which is what you'd expect. There's also the series base resistor - I think that's on the board too, as you'd expect.
I'd also expect to see a flyback diode across that relay coil - it's not shown on the schematic but I hope it's actually there on the board.
Unfortunately that conformal potting stuff on the board looks like it's going to make things horrible to work with.
If you disconnect the signal between the base resistor and the microcontroller and connect it to the +5V rail (the transistor side, that is, not the microcontroller) does the relay turn on?