I shoot with both hands and eyes, I started shooting left handed. This is with handgun, shotgun, and rifle.
The only adjustments I have had to make is adding an ambi safety.
It the 20 years of taking formal classes I have only had one fellow student in a class have to cover their dominate eye (tape inside their shooting glasses) this was because she could not close her dominate eye and shot with the other hand. Unless your son can not close his right eye and hold it shut while shooting you may not have to tape his glasses.
He will more than likely shoot better, not just because of his eye, but becasue he is changing working side of his body. It feels awkward, which forces you to concentrate on the basics (Grip, Stance, Sight Alignment Sight Picture, Trigger Control).
Practice 5 round dry, then 5 live rounds. I see many newer shooters at the range that have large groups; they are getting frustrated; I tell them the above after showing them how to dry practice. Their groups shrink rapidly.
It may also help that you start shooting right handed. Thus you are learning from the base up on a new skill too. It will help you figure out how to help him learn.