I got this quest since floor 3 and I am now on floor 9. The Maid Delinquent's continue to show and I capture every single one I see. Is it possible that I'm missing something or is this is a bug of some kind?

I think it it some kind of long term quest that you can do as long as you wish. It can be cancelled in the quest tab at any moment. 

It's a good way to add up the rep of the maid force, but adds a kind of enemy, so it's up to players to decide if they want to do this quest.

I have left it ongoing forever and honestly, I'm okay with it since the Maid Force loves me now.