Arduino - Ultrasonic Sensor for use instead of analog potentiometer for proximity control
The HCSR04 module is a digital proximity sensor, needing a microcontroller to read values from it. If you really want to use this then you will have to use a microcontroller as a digital to analog converter and map distance values from the HCSR04 to voltages.

However, a better solution would be to use some kind of analog distance sensor. I have used the Sharp IR distance sensors before (with Pololu carrier) and have had great success. This model also comes in a 3V flavor which might simplify your circuit design. You would now only need some analog circuitry to get the voltage values into the correct range.