The green light typically indicates that a GFCI receptacle is working properly. You can press the TEST button and the green light should go off, and then go back to green when you press RESET. But the GFCI function is electrical, not mechanical, so it has no effect on being able to push a plug into the socket.
If you have a TAMPER RESISTANT receptacle then it is designed so that you can only use it by pushing the prongs of the plug in at the same time and straight in. The mechanism to do this is mechanical and springs that control it can break or a piece of the shutter can get moved out of place or stuck. Those types of problems are often technically fixable, but probably not worth the effort.
If you do NOT have a tamper resistant receptacle and you can't plug something in then I would consider it broken beyond repair as that is simply not a normal situation. Something got bent out of shape or stuck and messing it with it is not worth the risk. 120V can kill quite easily. The GFCI will protect against current passing through a person, but it won't protect against a physical malfunction that could result in a fire.