I am pretty sure that most people can learn to shoot with weak eye and/or weak handed.
I am and always have been right handed and right eyed. Many years ago I got a bit bored with shooting and my performance was going downhill. For grins I decided to shoot left handed and left eyed. It was most weird but with persistance it came to feel normal and my shooting improved. I switched back and for a short while that too was weird but I quickly regained my former enthusiasm and skill. In my case I think that I had re-engaged my mind with shooting but for me at least it did prove that it is possibole to shoot well with weak hand and weak eye. Sure it is easier to use the strong side but one's natural tendencies can be overcame.
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TANSTAAFL