To get it you must collect all the dolls by killing bosses by nonlethal means. Do I have to collect them in a single run or do dolls collected in previous runs count too?
They carry over between runs, and I believe you can view them in Sunny's kitchen.
In answer to David McDavidson's follow-up question, in the Mission Briefings you are able to take control (prompts on-screen) of the Mk.II/III and navigate around the ship.
Simply travel upstairs (by pushing against the bottom step - Mk.II will automatically fly up) and in the kitchen, opposite where Sunny cooks her eggs, you'll see a bench. All dolls collected will be sat on the bench.