My 1999 Ford Explorer will turn over but it will not crank. We have checked fuel rail pressure and it has no pressure. Could the ignition relay or the coil relay be bad?

you might try starting fluid in air horn to see if it tries to run, then at least you would know it is not an ignition problem and only on the fuel side. Maybe wiring connector to crank sensor is bad. Really not sure how you would check if Anti-theft is preventing starting, but keys are only 1/2 of system and PCM is the other.

Did you check the fuel pump relay ? The relay gets power from a fuse and also from the PCM relay. The computer grounds the relay to turn it on for 3 seconds when you turn the key on, then only after the computer "sees" RPM or oil pressure.

It sounds like the pump is failing. That or sometimes not getting electricity. To check which it is, you could use a voltmeter or test light at the gas tank's electrical plug in. You need to check for power on the fuel pump wire when the engine is being cranked for starting. Do this when turning the key to on does not cause the pump to turn on for 2 seconds as it should. The fuel pump has power for those 2 seconds when key is turned to on, and again has power when the engine is cranked. You might miss the two second signal, so check for power when car won't start-when pump won't turn on-and with the engine cranking. If you have a signal then and still no pump coming on, then you must have a failing pump.
But if you have NO power to fuel pump wire with engine cranking, then it is a problem elsewhere-probably the pump relay (try a new relay if you haven't already). That relay is signaled by the engine computer when key is turned to on or when engine cranks . so you could do some testing at the relay if needed.

Check the valve on the fuel rail. Looks like an over sized tire valve. Push the center of the valve, if there is still pressure it will spray fuel. If it is still pressurized then you probably pulled the wrong relay.

Check for fuel pressure on the fuel rail while turning the key to start,if there is fuel pressure(enough)then the injectors are not pulsing(allowing fuel into the intake).The pick up coil in the distributor,will be the problem,unless it is distribuless ignition,then the cam sensor would be at fault.

Inertia switch is located Under right side of dash, on heater plenum. Do you hear the fuel pump turn on when turning ignition on? you can hear pump by opening fuel door and put ear against while having someone turn ignition on, another test would be to install a fuel pressure gauge on scradder valve and reading fuel pressure - recommended

is rotor turning?

If it has fuel pressure and fuel to the regulator and the rails, the regulator is working, it is getting spark to every cylinder and the timing is correct, You have good compression, it is likely a faulty sensor. You can test each one with a sensor tester or ohmmeter/multi-tester

Relay or sensor.

your fuel pump only runs for 2 seconds with the key on, unless it is running. Safety reasons. It runs during starting all the time, and all the time if engine is running.
check fuel pump relay in driver kick panel and inertia switch reset in trunk. Se owner manual for info.

you have pressure bleedng off due to a bad check valve in the fuel pump or a bad pressure regulator (on the fuel rail) letting pressure bleed off.

When you turn the key on & off, it starts the fuel pump for 2 seconds each time. The pump turns off after 2 seconds unless the engine is running. The pump runs all the time during crank mode.
But you need 40 psi when you start cranking so the injectors can spray fuel into the cylinders.
please rate as fixya if this identifies your issue. Good luck.