Does punBB use cookies for visitors?

Uses cookies to identify the logged in registered user.
A session cookie (which the browser deletes after closing the site page) can also be used if the redirect time is 0. Needed to send messages to the redirect page (pop-up message at the top of the forum page).

If the redirect time is greater than 0, then the engine does not set any cookies for guests.

> Am I right?

and Administration → Settings → Setup → Redirect wait = 1

then yes