Replace LED by a Trasmitter
I don't think enough information is provided for your specific use-case to give a comprehensive or definitive answer. I would be inclined to say yes; buy an existing transmitter, open it up and study it.

I would imagine a transmission is triggered by having the push-button close a circuit, allowing a current to flow. This could be replicated in your current circuit by replacing the LED with a transistor switching circuit, affecting the current across the push-button switch.

It would be much less tedious to use an existing transmitter/relay set than to try and implement your own