When I bought the car it had this symptom where when you turn the key to ignition, the fuel pump relay would buzz intermittently but not engage the fuel pump creating a no start condition. Although sometimes it would work fine, start and drive without problem. I swapped in a new relay, no change. Started looking at grounding issues and found the ground to the fire wall was disconnected. Cleaned and reattached the ground and it fixed the problem. Performed fine for about 2 weeks and now the problem is back 100% of the time. Checked grounds again, including the harness ground next to the drivers seat. No change. Checked voltage at the fuel pump relay and it oscillates between 8.5 and 10.5 volts. Any ideas what is going on here?

If you're not getting a full 12 volts at the relay it would seem to me that the issue is with the ignition switch. It's not uncommon to find some carbon build-up happening inside at the corresponding contact.
You can address this with a new ignition switch or, if you can, get inside and clean it up.

Thanks but the relay gets it's voltage from the fuse box and the ground from the ECU. I jumped the fuel pump relay to power on the fuel pump (which did run) and tried starting it but it still cranked without starting. I think there are other things controlled by the ECU that aren't operating as well...

Are you saying that you're getting 12 volts with the jumper in place of the relay?

According to the wiring diagram, when the relay is triggered, it supplies the ground to the fuel pump. So I jumped the relay, providing the ground to the fuel pump.

Have some corroded connections under the hood (sensors, etc). The terminals on the IAC valve were especially bad. Cleaned those and it cleared up the problem. Again. Probably going to have to go through all the connections on the engine wiring harness.

Well, the problem is back again. Checked the battery ground, cleaned the contacts on the IAC, wiring harness ground under the driver's seat. It's strange that once the car starts, the problem goes away. Going to try to diagnose the ECU. Seems to just be a starting mode problem

I suspect it is a grounding issue (almost seems like it's back feeding a ground through a sensor or something) so I pulled the ECU and checked that it had grounds to all the appropriate pins. Also noticed that the ECU will not go into diagnostic mode while the relay buzzing is occurring. When it does boot up, it was displaying an ac code and a speed sensor code but that shouldn't affect startup. Opened up the ECU and notices some burning on the board where it had looked like some components had been replaced. Thinking I might have a bad ECU (solder breaks)?