You can still get it. A copy of your DD Form 214 sent to the Awards and Decorations office in Alexandria should be all you need. That's assuming you were Army.
When I was getting ready to retire from the Army I had not received a GCM for my initial three years as an enlistee. They cut the orders and issued the award. I had not thought about it until then.
Search around the web for the Army Personnel Center, etc.
Years ago it was MILPERCEN. Something else now.
It's HRC now--Human Resources Command.
I served 2 years + 2 days in 1951-52. Nothing was ever said about "earning" a Good Conduct Medal. Somewhere I got the idea that 3 years service was required to get one. Actually I never got a discharge either, just a DD214. I suspect they may have me marked up AWOL from the inactive reserve, for which we were theoretically hooked for 5 years after discharge.