Wooden pegs breaking. Any suggestions  

Yes.
Better pegs.
The Authenticity Police will look the other way as you remove the original pegs, carefully noting the correct diameter and length before storing them in acid-fee packaging, and search for sturdier material -- not limited to wood -- from which to cut new pegs.
No ? Best of luck with your Cado Improvement Project (CIP) !

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SDR answered. Replacement pegs. I believe it is 1/4 inch, but not certain. Not really a DIY since you need clamps and such, but not a difficult project for a woodworker. The pegs are glued, not just 'pop outs'. They need to be drilled out at the correct angle. A solvent may loosen the glue but have not tried.
I have one unit by the garage entrance, so i bought some cheap dowels at the local hardware and cut them with a snap blade. Cut long, 4-6 inches, and have a half dozen in some the empty rail standard holes for keys, scarves, dog leashes...so i recall it was 1/4 inch.
I'm surprised it is happening to your shelf supports. I do have one heavy drawer unit where the back panel is wanting to pull away. Pegs are still very strong.
I recall reading once where the salesman in the NY showroom would climb the units to show how strong they are. Don't think i would try that today.

Hardware-store
dowels are made of softer woods today than in the past -- and in less-accurate diameters. I recently had need of a 5/8" dowel and found, among all the rubberwood or agathis (?), one heavy dark birch dowel. Hooray ! Who knows how it got into the store . . .

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Perhaps i was not clear Stephen, though you know me better than that, ; )
I used the cheap dowels for light duty keys and such. They fit snug whatever the diameter. It is standard though. I am off on travels and not home...
It is unusual for these dowels to weaken unless they were in a damp climate or
not tapped into the uprights properly. It is a system that works very well. No mater the vintage age of them.
A replacement dowel of a harder wood, and furniture dowels are made for such, would be a better replacement...
At such an angle that they are constructed, it does not seem possible they would give out so easily. If they are tapped into the wall standards properly.

Home Depot has oak dowels.
Just sayin.

All this peg talk
makes me really want to say something, but sadly I have nothing more to add to the conversation.