Servo controller boards

Let me apologize for my ignorance in all things electrical. My question is .After programming a controller hat to work 2 servo motors,can the aduino then be disconnected and the unit work as a stand alone,given the correct power source

Yes. Arduinos are generally programmed and then left to run standalone.

Thanks for the answer ,I think I phrased it wrongly , I wanted to know if the servo controller can be left as a standalone after programming

Will it have an Arduino controlling its input pins? Then the answer is still yes.