Oh boy, you are missing 4 chunky components
to the left of the power connectors. Check out Actually Hardcore Overclocking's video on the R9 390 Nitro. According to him, those are voltage regulators (4C10N ON Semiconductor) for VRAM *gulp*. Let's hope VRAMs didn't get zapped. This still doesn't explain why you don't have 12V on other voltage regulators tho. Did you check the fuses and 0 ohm resistors for continuity?
I would suggest you start by finding a high res image of both sides of the PCB and carefully check if there is anything else missing. I wouldn't be surprised if someone used it as a parts card (or you might be lucky and someone before you already found the issue in the missing parts and only removed those...).
From this point on I'll use the markings from the above video as descriptors. At this point, we are testing the power supply for the card so there's no need to plug the card into the PCI-E slot (I believe it doesn't use the slot voltage supply for any functionality to the right of the VRAM modules, but I might be wrong...).
1) You need to find out if the 12V rail is present at all the voltage regulators of the 6 phases of Vcore (this should be the case whenever the 8-pin plugs are connected) or there is continuity from the 8-pin plug to the VRMs. Each of the 8-pin plugs has 3 +12V rails which should go directly to the Vcore VRMs. If the voltage isn't present you need to check the fuses.
2) Find out if the VRMs (IR3550) are being supplied with a PWM signal by the 6+2 phase controller (IR3567B) as well as Vcc. If there is no Vcc there probably isn't a PWM signal either so we look into a logic voltage regulation part of the board under the 8-pin connectors. If there's no PWM signal but the Vcc is present we move to the phase controller. If the signal is present the core should be running (or at least have the voltage needed). You could check the voltage on the capacitors to the left of the Vcore VRMs.
3) Find the 5V and 3.3V supply. The VRMs use 5V and the phase controller uses 3.3V as Vcc so both should be present on caps around the corresponding chips. If one is missing find the voltage regulator for those and at that point, you've probably located the fault.
EDIT: I just noticed something isn't quite right with your C4139 (bottom left of the card next to the last VRAM chip)