Previous summer we sailed with a rabbit. I'm quite sure it doesn't work out that way, but my wife optimistically defined our ship as Norwegian territority, hence as long as the rabbit stayed on the ship we shouldn't need worrying about rules on taking pets across the borders. In any case, we've so far never been stopped by any custom controls.
It all depends on the countries that you sail to, sometimes there is no need for quarantine, shots and such. The paperwork also doesn't have to be extensive, it all depends. Just to be safe though, a visit to the vet's office isn't a bad idea. There are diseases dangerous to animals and not humans, local epidemics, air-borne diseases... Many things can happen which one cannot prepare for, but as they say "better safe than sorry", so it's good to keep yourself informed. Even if the pet doesn't leave the boat, they can still get ill or bring an illness with them to a new environment. Well, I hope you won't run into any trouble. A rabbit actually sounds like an excellent companion to have on a boat. Safe sailing to you!