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. 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. Alternatively you can just use 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.