Welcome to Electro-tech-online, Daisy!
This sounds like a great project! I just can't believe that the dog collar can't be adapted to activate a switch. Have you tried taking the whole thing apart yet? There must be some kind of switch already that triggers the output of the collar. You should be able to remove (or at least disconnect) this part and replace it with a relay. I'm not sure how they make these collars now. I've heard that some actually shock the dog, which would require a step-up transformer to increase the voltage. I've also heard that some just vibrate at a frequency that really disturbs the animal. In the transformer version, I expect the wires to the driver circuit of the transformer could be connected to the relay, instead. In the vibrating type, there's probably a motor. The wires could be removed from the motor terminals and connected to the relay.
Anyway, good luck and keep posting!
Der Strom