It looks like the point relays relax as soon as a score reel pulls in and then immediately reactivate. Is that about right? It could be that something is interrupting the power to the point relays, possibly because of the vibration of the score reel plunger pulling in.
I don't have a King Rex schematic but this is from Bowl O:
Bowl O point relays (resized).jpg
If the Anti Cheat relay switches in this schematic didn't stay closed the point relays might chatter as yours do.
I'd look around for sparks as edednedy suggested, especially on switches in the power rail circuit like those above. Also look to see if any other relays are chattering at the same time.
/Mark