If I'm reading my schematic right, pin 10 is the RPM input. It should not be a fixed 5 volts DC. If the relay does not see the engine rpm, it shuts the fuel pumps off. The rpm signal is generated by the crankshaft position sensor, and looks something like this on a lab 'scope:
It's an AC voltage that varies in amplitude and frequency as engine speed changes. The wide trace in the middle is what the ignition module looks for when firing the coil. In addition to the ignition module, the signal is also used by the tachometer, A/C compressor control unit, the CIS control module, and the fuel pump relay.
Since the car runs when it has gas, we can assume that the crank signal is OK when it gets to the ignition module. Somewhere between there and your fuel pump relay is where you'll find the problem.
The fuel pump relay pins are as follows:
1 Not used
2 Temperature input
3 Not used
4 Cold start injector
5 Not used
6 Transmission kickdown
7 Fuel pumps and O2 heater
8 B+ always
9 B+ in Start/Run
10 RPM input
11 Ground
12 B+ engine cranking