Forgot to add....
In my opinion the factory GM relay system with the oil pressure switch is a good design. It's a good simple, well thought out system. If my car had it, I would endeavor to repair and utilize it.
If the engine stalls in an accident the pump quits running. That's good.
yea like said that pressure cut off is good, How many wires are still disconnected try to google it and hook it up easy -safe.
Small update. First I jumpered the FP relay plug so it would provide constant power. Turns out it supplies power even with the key off, but that's ok for testing. The car ran fine for ~5 minutes without ever losing pressure, so the pump and circuitry seem to be fine.
Next I pulled the cover off the relay, plugged it in(making sure all-5 contacts were contacting the plug), and restarted the car. I heard the relay engage at key-on & the pump pressurize the lines, but after starting the car I had zero psi again, and could see the relay contacts were open. Manually closing the contacts instantly energized the pump & pressurized the lines.
Hmm, ok... I have no idea why previously the FP would run constantly, but suddenly now it starts behaving as it should per the manual, but let's run with that theory. After a bit of research online, I discovered what I thought was the oil pressure switch vs. the sender was completely backwards, so I disconnected my oil pressure gauge under the hood & screwed in the factory oil pressure switch.
When I started the car, rather than finding the FP relay contacts closed, I discovered oil spewing all over the back of the engine from the switch.
I checked it and made sure it was threaded in tight(it was), and did a bit of googling - turns out this is a known failure mode of the switch, so I have another one ordered and it should be here tomorrow AM.
Oh, and since the goal is for SWMBO to use the Vette to get to work if I need the van, or it's down for repairs, wiring the FP to a toggle switch is out(though it certainly was an option I've been entertaining this week). But the end goal is to have something she won't complain about driving too much, so I have to keep that in mind.
In reply to GTXVette:
Thanks for the call & message last night. I wasn't sure what your work schedule is now, so I didn't want to call back today if you were still working nights?
coexist
New Reader
5/20/17 10:05 p.m.
Toggle switch was suggested only for testing and narrowing down the problem area. I agree that the safety cutoffs for the fuel pump are a good idea and should be kept in place, once you know what is going on.
Regarding your relay, it's possible that the contacts were welded shut, so the pump would run without the signal power to the relay. It's also possible it unwelded itself and wouldn't run the pump at all. I would get a new relay, based on it's apparent behavior so far.
Thank you for being considerate of my sleep tme. I should have Told you tho... at 66 years old the 3pm till 2am shift was more than my body and brain could tolerate because I still wanted to get up at 6am and I am Kinda looking for something else. Kinda sorta.
IDK about Corvette in particular but I know 3rd gen F body fuel pump wiring isn't what one would think it is, the oil pressure switch is only there to bypass the relay entirely in the event the relay fails (if oil pressure is present the pump will run), the relay is normally engaged whenever the engine is spinning.
In reply to BrokenYugo:
Interesting... The oil pressure switch didn't arrive today, so hopefully it'll be in tomorrow, and I'll have time to install it & test it out tomorrow night. If it doesn't work I'll just tie the relay solenoid into a switched 12v circuit and go with that.
I may have nothing useful to add, but I have had a similar issue in the past with a fuel pump relay.
With my old Classic Saab 900 Turbo Loki, I spent hours chasing the fuel pump circuit. I had all sorts of issue getting the car to fire up and run. With the Bentley manual in hand I checked every single wire and connection in the whole fuel pump system and the relays. For a reason I never got to the bottom of the fuel pump just wouldn't run through the relay. I could run it through a separate switch that jumped the relay block, but replace that with a relay and it wouldn't run. I tried different relays. They all pinned out OK and bench tested OK. They all worked in different locations (the ECU uses the same type of relay) I had four of them. They all worked elsewhere, none would work on the fuel pump circuit. Every wire to that relay block I pinned out and checked not just for connectivity, but also for resistance. Everything on that damn car measured out OK, but it wouldn't fire the pump through the relay. My plan was to use a separate panel from Longacre or Summit or one of those are make a dedicated fuel pump circuit with a standalone relay, but I had a working session with other volunteers from here on the board helping and found the head gasket was gone so I gave up and punted on the project and sold it too EvanB who I hope will get it up and running soon.
I'm all for a stand alone wiring of the fuel pump, even if you add a new relay else where so your wife could drive it.
In reply to Adrian_Thompson:
Wow, that sounds quite perplexing!
I got a call confirming the oil pressure switch arrived. Hopefully SWMBO doesn't work too late tonight & I'll have time to install and test it.
Sensor is in. I like that they pre-coat the threads with sealant.

It still didn't fix the problem though. I check voltage on the FP relay and found the two terminals for the coil. I need to determine whether it's a switched ground or 12v circuit, then find an appropriate source I can jumper from.
I thought of one more test to try tonight - I unplugged the oil pressure switch and jumpered the terminals, but the FP still doesn't turn back on after the car starts.
Ovid_and_Flem had given me the contact info for a C4 guru, so I gave him a call this evening...and he's stumped too, which I both feel good & bad about: bad that I have such a perplexing situation for him, but also good knowing that I've been pretty thorough in testing & troubleshooting. He's going to do some research and get back with me, but in the mean time I'll start working on wiring up a toggle switch for the FP relay - but not tonight, the mosquitos were absolutely viscous after the last few days' rain!
NOHOME
PowerDork
5/23/17 7:50 p.m.
The golden rule in electrikery diagnostics: "If a circuit starts to behave like it is in need of an exorcist, then you have a grounding problem."
Stampie
SuperDork
5/23/17 8:40 p.m.
In reply to NOHOME:
In Air Force tech school I was taught the golden rule in electronics was never let the magic smoke out.
In reply to NOHOME:
Given there are relatively few grounds on a C4 due to the vast expanses of fiberglass it very well could be a ground issue. I've taken the time to clean each one I've found, but I'm sure there are more I'll never see.
In reply to Pete Gossett:
Look to see if one of the spade lugs in the relay's socket has broken loose--when you push the relay into the socket it would push the wire down instead of slipping on the tab of the relay--if this is happening, you would have contact by the spade lug just pressing against the tab until the socket moved when the motor started.
Alternate theory: wire corroded/broken inside the insulation where you can't see the separation, tests OK until the motor starts and moves the wire (I would call that a gremlin).
I'd try working backwards from the pump. Jumper 12v to the pump connector and hook up a test light at the relay end and start yanking on the harness until you're relatively sure there aren't connectivity issues in the harness itself. Then do the opposite for the grounds. Jump the pump ground connector to a known ground and hook the test light to the relay connector and 12v and start shaking stuff again.
In reply to ITBland:
There's definitely signs of the connector nearing failure, but I checked for voltage from the underside of the connector directly on the terminals, and still didn't have voltage there.
I finally had time to install the toggle switch today. Initially I was going to get all fancy and route it into the interior, but that seemed like quite a bit of hassle for what(at this moment anyway) is a temporary bypass for troubleshooting. So I tapped it into the 12v in/out wires that run through the relay -> fuse -> fuel pump, and Velcroed the switch next to the relay.
A couple days ago when I was troubleshooting it, I jumpered these same wires through the connector with a spare piece of wire, and discovered the FP would run even with the key off. Today after wiring the switch so it completes the same circuit, the FP did not run with the key off. Not only that, the FP continued to run after startup with the new switch off.
berkeleying flakey 30-year old GM E36 M3... 
The good news is the car idled fine for 20-minutes. I also put it into reverse, then drive & back to park without any trans fluid leakage. So I'm planning on calling to get the insurance started on it tomorrow, then hopefully tomorrow evening(if it starts) I'll pull it down the drive & back to confirm the trans cooler line is still not leaking. After that I need to bolt the AC drier back into place, and possibly take it about 1/2-block down the street and back.
So Like YEA, Have you compleated a Break in on the Cam ?
I'm wondering what the underside of the fuse block looks like. I had a bunch of weird stuff happening with the 911 for a while after I got it. I disassembled the fuse box, took everything out, cleaned every contact, fuse, and relay, put dielectric grease on all of them, reassembled, cleaned every ground in the car (there are a LOT in my 911) and it fixed every electrical issue.
In reply to GTXVette:
I'd say it has an hour+ plus of combined run time now, so it should be broken in, correct?
In reply to mazdeuce:
I cleaned all the terminals in the fuse box. They had white crud that I'm guessing was the remnants of prior dielectric grease, but weren't corroded. Grounds are a known issue on C4s too. I've cleaned all the ones I've encountered, particularly the ones from the battery & engine, though there are dozens.
I can't be the only person reading along thinking "man, if I just had three or four hours to crawl over the whole car and wiggle/clean/tighten stuff, I could totally fix it" even knowing that you've already done that.
I was trying to think earier how I could get down there for a long weekend and help. so, no, youre not alone seth.
In reply to Dusterbd13 and
mazdeuce:
You guys are awesome for even considering that! 
Yo Dustbuster, Michael If you ever do that road trip Give me a holler.Not much on my schedule that can't be moved for a day or two.