One of the first things the Army Marksmanship Unit does with new members,  who are all already accomplished shooters,  is an eye dominance test,  and retrains them to shoot with their dominant eye if they've been shooting cross- dominant,  and they've always seen an improvement in scores.   Shooting is more mental than physical strength.

OP - if you've got a rifle with a rail,  get a MantisX and a laser trainer cartridge,  and have him dry fire with both.  The laser will show where the muzzle was pointing when the hammer fell,  the MantisX will tell about movement before and after the shot.  Teach him about how to build a solid supported prone or bench rest position,  sight alignment,  and trigger control.

Cross dominant shooters is nothing new, and can be easily trained.  He will need to practice, so look at it like playing a musical instrument in that you don't play a concert in public after 3 lessons, it takes practice with repetitions (Meaning the more you do it, the more it becomes second nature).

Being that you don't have a range, you have a couple of options.  Is there any Youth Air Rifle clubs around you, I see that you are in Texas so you may consider contacting https://tsra.com/youth-programs/ and see what they have to offer, as a club will usually have coaches and can be a great help.

If you wish to do this at home, then consider doing pellet/BB gun work, as you can find pellet/BB traps (back stops) at many of the sporting goods stores (Academy, Cabelas, even Walmart) or make one yourself, and these can be indoors/outside.  Now consider the type of rifle he will be using, at most Scout Camps they are using a single shot .22LR bolt action with iron sights (Generally peep sights, though I have seen others), so for finding something similar to work with you can find pellet or BB rifles that have peep sights.

Focus on fundamentals of where he places his head on the stock (cheek weld) so it is consistent to get the same sight picture every time, then focus on obtaining a good sight picture (sight alignment & sight placement on the target), breathing, trigger press.  Do you know if they allow the Scouts to brace the forend on anything at the range like a block of wood, if so then get him something similar and work on that as well.  If the Scout Camp range doesn't have something for the Scouts to brace on, then work on having him place his hand where he can brace to give the best support and steady the rifle, and work at being consistent.  Do this 30-60 minutes or more at a time, several times a week for several weeks or months, and he will to the point it becomes more comfortable and natural, we do this same type of repetition training for holster work and combat shooting training and with lots of repetitions it becomes second nature.


CY6
Greg Sullivan "Sully"
SLR15 Rifles
TheDefensiveEdge.com
(763) 712-0123