2003 Dodge Caravan 3.3 No Crank, No Start

The most likely reason would be bad contacts on the starter relay or starter solenoid. You aren't getting enough amps to the starter.

https://shop.advanceautoparts.com/find/dodge-grand-caravan-starter-solenoid

May need to check fuses and relays

85 & 86 are the control side of the relay , one should have battery voltage (85) the other (86) doesn't .86 is control from the PCM - engine computer . PCM supplys a ground to close contacts inside the relay sending battery voltage to the S terminal on the starter solenoid . If the PCM doesn't see a voltage signal fron the park neutral safety switch it won't energize the relay .
Circuit Description (ECM/PCM Controlled)
Moving the ignition switch to the START position sends a 12-volt signal to the engine control module (ECM)/powertrain control module (PCM) Crank Request circuit. The ECM/PCM verifies that the transmission is in the PARK or NEUTRAL position. With the transmission in Park, voltage flows through the Park/Neutral position switch and feeds the starter relay coil. The PCM then grounds the control circuit of the starter relay. When the starter relay is energized it allows battery positive voltage to the starter solenoid S terminal. On some vehicles the starter will continue to crank the engine with the key released until it starts or the crank command has timed out to prevent excessive heat build up in the starter circuitry or the ECM/PCM receives an engine run flag.

The ground is aso controlled based on the neutral safety switch input

could have bad battery cables.negative battery ground cable could be faulty.check starter 40 AMP fuse or look for bad starter relay.

Yes there is a starter relay in the wiring diagram.
The yellow and lite blue wire goes from the starter solenoid to the relay. There is a yellow wire from the relay to a 30amp fuse. The key switch powers thru the neutral safety switch and then powers the relay on.

Check the neutral safety switch on the transmission. It should have three wires on the connector (if I remember correctly). There should be power on the Brown with yellow wire with the key in the crank position. If there is ground that wire and see if the van starts. If it does replace the switch. If it does not check the starter relay. Let me know if you have any questions and I'll do my best to answer them. Thank you for using FixYa.com!

I’m happy to assist further over the phone at https://www.6ya.com/expert/jeremy_d728a59f986299fa

It's most likely the neutral safety switch. Try putting the shifter in Neutral and slipping it just a little toward Park and see if it'll crank. If it does the neutral safety switch is the problem but worn linkage is the cause.
You've already check or replace everything else so it has to be this or the ignition switch.

there is an interrupt switch, called neutral safety switch. When it fails or goes out of adjustment, no start, and essentially no protest or indication. Try foot on brake, shift to neutral, turn key to start and slowly move shifter out of nuetral in each direction. If starter actuates, look into switch adjustment.

This is a bad PCM. Try to disconnect the security system first and if it still does not start then it is the PCM.