There are plenty of toys that make use of wireless remote control, usually these would be some sort of toy cars. If one wants to design such a device there would be a microcontroller in there somewhere. What ICs can be used for the wireless data IO (i.e wireless trasceiver ICs suitable for such projects). Does one have to use a specific antenna type?
Find out about frequency band allocation for radio control. You should operate on a legal frequency.
Your transmitter must translate your commands into an AM or FM broadcast. A train of commands might normally go out 50 times a second. The antenna can be ordinary rod or telescope, a few feet long.
Your receiver must be good at picking up radio waves and amplifying them. The antenna can be a rod or length of wire a foot long. Your receiver then must extract the commands in the broadcast.
You need an wifi module embed toToy. Chipset you can choose Ralink5350, Realtek8196, or Athores9331, Athores9344 etc, depends on the features you need.