From my bench file dated 2005:
Blue is +12 Volts. A close look should reveal that it connects to the right-hand leg of a 78L05 chip.
Gray is ground.
White is audio.
Brown is the activated line. The toy plays while this wire is grounded. A pushbutton switch to ground will do. Pretty sure they mean it to also work connected to the radio's transmit pin of the mike socket. All radios are not created equal in this department, but it should not damage the radio to hook this wire to the mike socket's transmit pin.
You can make this toy edge-triggered rather than level-triggered. The way it's built the sound stops when you break the ground connection to the brown wire. Only plays while the switch is held closed.
To make it start playing when the switch is closed, and finish the whole sound clip if the switch is released:
Cut the trace between pin 23 and the diode connected to pin 23. Jumper the anode end (not banded end) to pin 24. Pin 24 should already have a 100k resistor connected there, and to 5 Volts at the other end. They call this a "pull up" resistor.
73