If your joystick comes with software that allows you to map that mini-joystick in such a manner, that'd likely be the best option.
The DirectInput interface used for joysticks does not allow the manipulation of existing devices, so while manufacturers can use a custom driver that's specifically designed to circumvent that limitation, software like VoiceAttack cannot press buttons on your joystick.
If you're looking to trigger VoiceAttack commands (that send keypresses, for example), that could be done using a command that runs a loop checking the position of the axes of your mini-joystick, and performs a given action when the value exceeds a certain threshold.