Here is a method (Sandy Wylie's) to correct blinking. You have to relax the shooter to the point where she can keep relaxed and absorb the visual and physical input from the gun. This method is the short route; the real answer is a Zen-type awareness.
If you have a safe berm that you can get close to, get within 5 yards. You want to shoot into the berm without a formal target to get comfortable with the gun. If you shoot iron sights, try just looking over the top of the gun instead of at the sights. Wear plugs and muffs to reduce the noise problem. You might find it of benefit to start with a .22 as well. To help in keeping relaxed, try to keep your facial muscles relaxed. Monitor this closely. I work on relaxing the muscles behind my ears for best relaxation and awareness. Use a relaxed grip on the gun and eventually work with weak and strong hand shooting. Repeat until you are comfortable keeping your eyes open while looking over the sights, firing downrange into the berm without a specific target.
Once you can keep your eyes open for the complete cycle, start watching the sights and monitoring yourself closely. If you are blinking, go back to no target and looking over the gun. After you have achieved the ability to keep your eyes open and relax, you will find a tremendous increase in awareness.
Credits to: Sandy Wylie.