The effect of “Lightning Punisher” is a Trigger Effect. So it will start a new chain itself.
“Lightning Punisher” must have been on the field before the chain began and must have been on the field throughout the entire chain in order for its effect to activate.
The effect of “Lightning Punisher” targets.
The effect of “Lightning Punisher” will still activate even if some steps of the chain have their activation negated. (Or even if “Vanity’s Call” negates all Links in the chain.)