A Puppet with this ability cannot fall asleep. If they try using the move Rest, the move will fail and the turn spent using it is wasted. If a sleeping Puppet gains Insomnia through Skill Swap, then they will immediately wake up. If a Puppet with Insomnia loses it through Skill Swap, falls asleep, and re-gains Insomnia by switching out, then they will immediately wake up.
Vital Spirit is identical.