Pump replacement thinking would come after proving that there was a an electrical supply available to the connector at or near the tank...? Sorry, I have no idea of the set-up on your fuel pump control circuit but generally other than anti theft aspects to the circuit which you seem to have an indicator for that determines all is well...
The fuel pump circuit is likely to have a safety circuit that would stop it pumping fuel out of a broken fuel line in the case of an accident where the driver may be incapacitated and the ign still on. In an attempt to achieve this some sort of switching system may be built into the fuel pump circuit that will only be closed when the engine is running.
As I suggested earlier, my 86 Morgan uses an oil pressure switch to determine that the engine is running, if there is no oil pressure the supply to the fuel pump is cut. Logically there is no oil pressure when you turn the ign key to start the engine, in which case there is a by-pass switching action built in... that only by-passes the need for the oil pressure switch to be closed to feed the pump... When the ign switch is held in the start position. Your system is unlikely to use oil pressure as does mine, however you can perhaps see the logic of the operation..?
I suspect there are many ways for a digital system to determine that the engine is running, using crank position sensor or some such... but ultimately it would again seem logical that if there is a fuel pump relay then it would be the final link in the system which if energised (clicks) when the ign key is turned to START, at that stage the pump should receive an electrical supply, and if the engine then fires up the engine running aspect of the pump circuit should logically take over supplying the link to hold the fuel pump relay contacts closed...
To be sure you are looking at the correct relay, and IF the wiring conforms to the colour codes, there will hopefully be found a white/pink wire leaving the relay on it`s way to the pump...?
Again the relay only energised either when the ign key is turned to START, or the engine RUN circuit is activated.
And as Paul suggests even anew relay can be faulty these days thus even if a relay clicks that does not indicate that it`s internal electrical switch is providing supply to the pump a voltage check is required to prove that it is working, of course if the pump can be heard running then no voltage check required
Last thought... is the fuel gauge operational...? On the off chance that both the fuel gauge and the fuel pump use the same earth wire at the tank connector and it has suffered some sort of corrosion at the connection or wherever it pucks up it`s earth connection to the chassis... worth checking you have a good earth..?
Hope this helps and does not further muddy the water for you.
Being an old Luddite, my thinking could be entirely flawed
