+1 on the battery connection, from your description, "moving it out of the way", I assume it doesn't get driven every day, so corroded terminals are a good bet.
There are actually 2 clicks when key is first turned on, but they happen almost at the same time.
On the passenger side engine bay there is the fuse box, under that are 3 relays, EEC relay, Fuel Pump relay and WOT(wide open throttle) relay.
EEC relay activates with the key on(click), it powers the Fuel injectors, spark, fuel pump relay(not fuel pump), and of course the EEC(computer)
When computer(EEC) gets power the first thing it does is to Ground the Fuel Pump relay for 2 seconds, the second "click", to prime the fuel system.
So second click is almost simultaneous.
In the fuse box there are a few 30amp fuses, one is for the Fuel Pump, check it first.
You can manually test fuel pump relay using the OBD1 test connector.
Look here:
http://www.therangerstation.com/tech_library/OBD_I.shtml
OBD1 connector is usually on drivers side of engine bay near firewall, grey or black plastic, it is loose not attached to anything.
Second diagram in above link has labeled slots in the test connector
Fuel Pump slot is the Ground for the fuel pump relay
Turn the key on
Then use a jumper wire in that slot to a Ground, battery ground is fine or engine, or ??, as long as it is a good ground then you should hear the Relay "click" and the fuel pump should come on.
If you don't hear the "click" then you will need to remove fuse box and get to the Fuel pump relay.
Easier place to test if fuel pump is being sent power is at the inertia switch, it is in passenger side of cab in the kick panel or just above carpet under glove box.
It is a small box(2"x2") with a button on the top, 2 wire connections, inertia switch is there to cut fuel pump power in case of a roll over or accident(sudden stop), button on the top is a reset in case it is accidentally tripped.
It is a 12volt in and 12volt out pass thru
If fuel pump relay is working then either terminal on this switch should show 12volts for 2 seconds when key is first turned on.
Fuel pump power runs like this:
Battery--------30amp fuse----------fuel pump relay-------------inertia switch---------------fuel pump------Ground
Ground wire for fuel pump is usually located near inertia switch, usually.
Just for FYI, the fuel gauge sends a 5volt pulse, yellow/white strip wire, to sender in the tank, so it won't show voltage on meter, and sender usually has it's own Ground, orange wire, so there are 4 wires into fuel tank