Hi Mark, I'm Clayso a certified master automobile technician with 30 years of automotive repair and diagnostic experience and I'm here to help with this problem. Additional services may be offered for detailed information.
Glad to see you are using a test light for this circuit. On fuel pump circuits I normally use a sealed low beam headlamp to verify the circuits integrity. I use the lamp just like a test light. Ground one side and verify power with the other. Low amp test lights and voltmeters will show power, but not under a load.
From your description it sounds like there are a couple of things going on. Check the ground at the thermostat housing, make sure it's there and it's not corroded and the connection is clean and tight. The ground on the thermostat housing should be a black wire with a white stripe, this is the same blk/wht wire on the fuel pump switch aka. fuel pump relay. If this ground is bad the relay will not close/click. To check the fuel pump relay itself, swap it out with the horn relay and see if the horn still works. The relays are the same.
The four wires on the fuel pump relay should be:
Dark Green/White == Fuel pump relay control, from the ECM
Black/White == Relay control ground
Orange == Power from gray fusible link
Gray == Power to fuel pump
With the relay disconnected you should now only have power on two connections. The dk grn/wht with the key on and orange should be hot at all times. The orange wire is power from the gray fusible link on the passenger side firewall. The link below is the fusible links, fuel pump relay and an inline fuse to the fuel pump. (When the key is turned on the inline fuse should power the fuel pump when the relay clicks). Check these fusible links and make sure they are not rubbery feeling.
Inline Fuse & Fusible Links
A fuse link is a short length of wire, integral with the engine
compartment wiring harness and should not be confused with standard wire. The fusible link wire gauge is smaller than the circuit which it protects. Fusible link wire is covered with a special thick, non-flammable insulation. An overload condition causes the insulation to blister. If the overall condition continues, the wire will melt. To check a fusible link, look for blistering insulation. If the insulation is okay, pull gently on the wire. If the fusible link stretches, the wire has melted.
I can also offer additional services for 5 bucks and continue this conversation on the phone and or through private email with diagrams and technical information specific to this vehicle. Let me know if you're interested.
I hope this information is helpful to you.
Should you have any questions please let me know.
Positive ratings are greatly appreciated.
Thanks so much.