Upon turning in quest and accepting reward, the quest icon appears again and one can repeat the quest over and over, although you don't get a repeat of the reward. I've "completed" this quest about 10 times in a row now without having to gather the papers again. Might be a bug.
Right, I have multiple pets and it's letting me repeat this quest over and over and over and over and giving me the completion screen too.
This is how the quest works, once you have completed it you can do it over and over again, with the intention that you use it on new pets.