How to remotely control a remote control over the Internet?

I recently installed a webcam in one room of my house to check on my cats when traveling, which works great except when its dark. It just so happens that the ceiling light in that room has an on/off remote control. Since I have a wireless LAN network in the house, I'm wondering if there's any device available that would allow me to remotely control that remote control to turn it on/off in order to illuminate the room.