It could be your starter relay or your starter is getting hot and cutting out. I would jump the relay and see if it starts while it is at operating temperature. If it does start, then replace the relay.

 

could it be vapor lock?

 

Belvedere1965

Any update on your issue, were you able to fix it? was it the relay or vapor lock? Just curious as my Fury does that sometimes, starts fine the first time, but is harder to start (but will start) when hot, unless it sits for 20 min or so, then it starts just fine!

 

Vapor lock is gas boiling and turning to vapor in the lines/filter/bowls. Won't affect the starter.

 

Try some header wrap around your starter, sounds like heat bog to me.

 

It could be a number of things.

But, since it sounds like the starter is functioning properly. The original post says it's NOT clicking. That is the sound of a dead battery.
It seems the engine is turning over just not starting up.

We need to focus on electrical igintion and fuel delivery.
When the engine acts up like this again, grab a spare spark plug or take one out of the cylinder head and plug it into one of the plug wires. Remember to put the spark plug next to a thick metal object like your exhaust manifold so you can create an arc.
With a buddy trying to start your engine watch the spark plug.
If you see no sparking, you have an electrical problem.
If you see spark, you have a fuel problem.

Let us know what you have and we can go from there...
Good luck.
Mopar_71