91 RS 305 TBI/auto. Have had this problem since I bought the car 5 years ago. Fuel pump will NOT prime/run at all until engine is already turning over (I'm guessing it's only starting because the oil pressure sensor is seeing oil pressure), car usually takes 5 seconds of cranking to actually start when it's cold. Have checked timing, checked all fuses including the ECM/Fuel Pump one near the battery, tried swapping fuel pump relays, and also checked wiring at the fuel pump relay: orange (always hot) wire shows 12V, Black/White (ground) wire is good, Green/White (ECM) wire shows 12V for 2 seconds when key is on, but jumping Orange (always hot) to Gray (fuel pump power) does NOT do anything at all. Connections at C313 above the rear axle look fine, and if apply 12V to the fuel pump directly, the fuel pump comes right on. Fuel pressure is 13 psi when it's running, and car seems to run just fine after it starts.

Can anyone think of a reason that I'm *only* getting power to the pump when it's cranking?

 

I don't have a wire diagram, so I can't verify colors. However, from your description the computer is trying to prime the pump.
Clearly, there's an open in the wire harness in the gray wire.
So now it's time to trace the gray wire.

I expect the break is ahead of where the oil pressure bypass wiring splices into the gray wire. (Almost certainly in the engine compartment somewhere, not back by the rear axle.) The Genuine GM service manual would have a diagram and parts location information.

 

I had the same problem. The wires and contacts on the fuel pump relay connector were fouled up with grease and dirt. It also looked like one of them pulled out of the connector a little bit. I cleaned them and pushed them all back into the connector very firmly and it started right up.

 

I think my problem is elsewhere -- likely the check valve in the fuel pump. I finally got someone to turn the key on for me when I'm standing near the fuel tank, and have verified that my pump does indeed pulse for 2 seconds when the key is on, I've just never been able to hear it over the key buzzer until I had someone else do it for me.

At any rate, I've rewired it so that my fuel pump runs all the time when the key is on. Car will now crank right up when cold, as long as I leave the key on for 3-4 minutes first to build fuel pressure. Going to go with this until the fuel pump quits completely.

 

The pump runs but doesn't build pressure during the two-second prime? That's not the check valve.

IF (big IF) the pump is getting proper voltage/current--and that's not guaranteed--the pump is failing, or the rubber hose connecting the output of the pump to the fuel hanger has ruptured, or the fuel filter has almost completely plugged. Get prepared to replace stuff. You need a pump (or at least the hose/clamps, or a filter) NOW.

If the hose is ruptured, a replacement hose has to be rated for submerged-in-fuel use, not all "fuel injection" hose is.

If the filter is plugged, that's really hard on the pump--expect the pump to have a shortened service life.

Re-wiring the pump to run with the key on is not a reasonable work-around, and it's not safe. In a collision, you might be knocked unconscious. The pump will keep running--feeding the fire at the fuel leak--until the fire department pulls your corpse out of the car. The OEM wiring shuts the pump off if the engine stalls.