One of the Buttercup toys can be found after the Lone Wanderer has both visited a few of the other locations connected to the engineering core, and possibly destroyed one or more generators. It appears on the bottom floor of the engineering core, on a table near the makeshift bed where Toshiro Kago usually sits.
A second Buttercup toy is also in the engineering core, on the table next to the pre-War bed near the broken teleporter that will later lead to the observation deck. The table is always there but the toy only gets placed on it when three generators are destroyed.
A third can be found on the shelf above the unique atomic pulverizer behind a lock (75 Science skill) in the weapons lab, that can be alternatively reached through the console beside the door.
A fourth can be found inside the research lab. Find the room with full-size buttercups on giant shelves. From the top lookout, looking over the shelves/rows of life-size Buttercups, you can see a small toy-size buttercup near the top shelf. Shoot it off and then go retrieve it or jump and grab at the right moment.