I tried starting my 1986 after my engine swap and at first it sounded like it would kick over, but it didn't and now the engine doesn't make any noise and all I hear is a fast clicking noise (sounds like it's by the starter), but last night the car fired up fine, please help!

 

The clicking is the solenoid at the starter trying to turn it over.
Check all your wiring. Also make sure your battery is charged.
You may need a new starter regardless if it just started last night.

 

I have a 1991 Z28 5.7, lately I have not been able to get the car started, I have replaced the fuel pump and the starter, both are working fine, I have spark, car still will not turn on. The key will activate accessories but will not turn the motor therefore I ran a push button to bypass the key. This still does not work, I am starting to think that the VATS may be the problem but am not sure where it is located or how to test it, can anyone help?

 

I had the exact same problem when I did my swap. Checked battery starter, and all that other stuff. The problem turned out to be the grounding strap that goes from the firewall to the back of the passenger side head. Weird, I know, but that was the problem.
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I think you should get your own thread next time but as for your problem, I think when VATS is involved, the engine will crank but not start. You indicated the key will not turn the motor, which leads me to believe it is not VATS related. Also, VATS related problems will usually shoot you a trouble code as well. Have you tried pulling trouble codes?