Yes, that would be a problem
Fuel pump(FP) relay has the Green Base/socket
EEC relay has Brown Base/socket
If key is off and it does this then there are two problems.
The FP Relay's coil only has power when key is on, but it has no Ground so shouldn't activate.
Ground is provided by the computer OR VIP(OBD1) connector.
Power for the FP Relays coil comes from EEC relay, so it maybe stuck closed
Battery------EEC Fuse-------EEC relay-----------FP relay coil
EEC relay also powers fuel injectors, spark system and the EEC(computer).
FP relay coil-----------Computer-----Ground
FP relay coil-----------VIP(OBD1)
OBD1 connector is in the engine bay it is part of the main wiring harness and is hanging loose.
Look here for diagram of it: Ford Ranger - Testing EEC-IV Equipped Engines
The fuel pump Ground slot is labelled in that drawing, if key is on and you were to Ground that slot with a jumper wire the FR relay should "click" closed and fuel pump should start running.
Wiring diagram here: http://www.therangerstation.com/tech...0_3_0_1of3.JPG