This quest appears after completing the quest Collect 'Em All: Strangers from the Outside!. The time between completing the quest and this quest being offered is just over two days.
The Alien Toys required for this quest appear inside the Dream Castle and Remy's Restaurant. The first 8 toys will be available to find after accepting the quest, and about 36 hours later the remaining 7 will appear.
Initially 4 toys can be found in the castle, and 4 inside the restaurant. Then about 36 hours later the rest will appear in the castle and restaurant as well.
Quest Text
Buzz seems to have found more of the Alien Toys. You'll need to speak to him.