I know that in the British Army at the time, commissions had to be purchased for a given sum (usually from a retiring or promoted officer) but what about the Continental army?
Were they issued by Generals or did they have to come from the Continental Congress? Also, were they given to anyone, or were they only issued to what would be considered a "Gentleman"?
Most of the time it was controlled by the state's legislatures. Each state had a number of units (or a "Line") that it was supposed to contribute to the main Continental Army in addition to the militia and long service units meant for local defense. Thus it came down to influence and political connection in many cases. Congress did raise a few regiments directly, along with the artillery and corps of engineers which they could directly appoint officers to. Then there were a few units raised by individuals, such as the legion raised by Light Horse Harry Lee where the leader could obviously appoint his own officers.
The militia regiments often did elect their officers, particularly in the NE units and earlier in the war when many were formed quickly.
Washington many times was deeply troubled by the quality of the officer corps in the army. He thought they too often were too familiar with the men and not of "proper stock" which he thought important. The theory being that men who were born to that class would be more naturally able to command, along with bringing dignity and professionalism to the new service.
Great source: Patriot Battles, How the War of Independence was fought by Michael Stephenson