The vet is right about the possibility of genetic conditions coming up.
Pedigrees don't even tell you that, if the chins have genetic conditions. A reliable and knowledgeable breeder you can trust will tell you that.
Pedigrees are good for being able to trace lineage, knowing what genes are present in the lines as far as recessives or dominant, etc... (say you have a standard for example - but seven generations back there is an ebony mentioned in the pedigree...and one day you have an ebony born from that standard chinchilla. That would explain it. As ebony can pop up generations later in the lines.) and of course knowing the background to prevent breeding siblings or relations together.
There are issues to worry about like malocclusion or congenital heart defects, that you might not be aware of, but your chinchilla may carry these genes (with maloccluders, sometimes they don't even present the malo until they are seven years old or older.). All it takes is for you to breed one chin carrying malo with another who carries the gene, and one in every four kits would be maloccluders. 2 in every 4 kits would carry the gene for it (which if they were bred, it would continue an endless cycle of maloccluders if bred to others.). If you have an actual maloccluder, which you may not find out till they are much older in age, then every kit out of that maloccluder will carry the gene. If you have a maloccluder and breed them with a chin who just carries the gene, then you have a chance of either kits born carrying the gene for malo or kits born that will express genetic malo later in life.
Malocclusion is proven to be recessive which means it can carry for generations too, in a line and resurface later. All it takes is for two malocclusion carriers to be bred together to make it happen.
Fur chewing is another issue, that is genetic and may present itself in the animal at an early age or much later in age. As far as breeding chins from a pet store, I wouldn't recommend it, since you really don't know what you are breeding for or going to get. Imagine the heart ache of losing your chin to malo at a young age, and then think if you were to sell kits like that to others, how it would feel? It is just to me, an endless cycle of pain and suffering.
There is a lot more to say when it comes to breeding without a pedigree, but I do not want to write a novel! My suggestion, if you choose to breed, is to hook up with a breeder near you that is reputable. They can mentor you and trust me, breeders are more than happy to help new breeders.
As far as even thinking of breeding a pet store chin... I had a couple over yesterday who had a pet store chin they had bought back in December. Poor baby had wet eyes, and when I checked his mouth, I found spurs on his lower teeth.
I was sorry to have to tell them that, and I believe they will be scheduling with a vet for an x ray soon. It made me sad to see the owner cry. Her chinchilla is really beautiful, really beautiful (reverse mosaic), but sadly you can't sugar coat or fake when it comes to tooth troubles in chinchillas I am hoping somehow it isn't malo, but I don't have a good feeling with the chin's eyes and teeth being that way.
They had thought about breeding too, but I am so glad they chose to keep him as a pet.
As I said - sometimes a little bit of stress will bring it out - in reference to chinchillas who weren't chewing and then suddenly do chew. 