No, not directly at least, microphone's don't output any signal without power going to them. Typical microphones are electret mics, AKA piezo microphones. They require 2-9 volts of DC current applied to them and a resistor to bias the built in FET amplifier, the signal you get from this is still pretty weak so it will need to go through a pre-amp to give you an output signal that's compatible with a line output (headphones) Since the microphone requires a DC bias for power AC coupling is required. You could use MANY simple audio mixing hardware to feed the RF adapter a signal from a microphone though..
One possible source for this type of audio equipment on the cheap is a local Salvation Army or any similar type of thrift store, many people offload older AV equipment, and you'll find quality stuff if you look.
Another easy solution is to use your PC, they almost always have a mic input, and through the mixer in the audio control panel you can cause it to output the microphone to the speakers, which you would just replace with your RF adapter.