I have any grounding electrode sysytem that has current flow of about 1.5 - 2 amps, but I have no voltage on the grounding electrode wire. How can I be reading current flow? Where might it be coming from?
The types and sizes of conductors used as grounding electrode conductors have very little resistance. If we picked, just as an example, a resistance of 0.02 ohms (a reasonable value, for illustration purposes), a current of 2 amps would be created by a voltage source of 0.04 volts. Is your measuring device capable of detecting that small a value? While we are on the subject, what type of meter are you using to measure current? Depending on the type of meter, you could be reading "cosmic radiation" or some other phenomenon that does not represent actual current in the wire.
This is quite normal and expected. It is a result of parallel current flowing back to the transformer via earth. Earth and your grounded circuit conductor are in parallel. Even though the grounded circuit conductor has extremely low resistance compared to your ground electrode, current will take all paths available to it.
What is driving it is the voltage drop on you grounded circuit conductor between the transformer and your service disconnect at the Main Bonding Jumper. You will not be able to measure it unless you have some extremely long leads to reach those two points and a bucket truck to get up to the transformer which would be against the law.
What you need to know is how much current is actually be carried on the Grounded Circuit Conductor by measuring it and comparing to what you see on the GEC to see if there is a real problem or not. For example if you have 10-amps of current on the Grounded Circuit Conductor and 2-amps on the GEC, you have a problem. That problem would be your Grounded Circuit Conductor is opening up somewhere like a splice or connection, or you have an extremely low ground resistance which is highly unlikely.