They do not respawn after initially spawning. Vindicators generated in mansions do not naturally despawn unless the world is switched to peaceful difficulty‌ ^{[JE only]}.

What happens if you name a Vindicator Johnny?

It is that when a Vindicator is renamed “Johnny” with a name tag, it will become hostile towards every mob, but they will become passive towards you and all the other players. They will not attack players but will attack other mob when renamed as “Johnny”.

Do Woodland mansion Illagers Respawn?

Illagers do not respawn, so once you’ve killed them all they’re gone for good from that mansion. Zombies, creepers, skeletons, spiders, witches, and endermen will spawn naturally per normal rules.