Can I sue a puppy seller for selling an inbread dog that had to be put to sleep due to complications?

If the seller knowingly misrepresented (or lied about) the breeding, pedigree, and/or health of the dog, that could constitute fraud; fraud provides a basis to either rescind the agreement (get your money back) and/or sue for damages (monetary compensation). One key issue is generally whether it was a knowing misrepresentation, or a lie which the person knew was a lie; if the seller him/herself did not know of any flaws in the health or breading or pedigree, there's a good chance he or she would not be liable. In this context, "knowing" generally includes when the person reasonably "should" have known--that is, for example, that any reasonable breeder (if a professional breeder) or at least dog owner (if a private seller without any particular experience or training) would have known of the problems.

You'd also need to be able to causally link the breeding, etc. defects to the health problems; if the health problems were unconnected, that could reduce or even eliminate what you would be entitled to. For example, say the dog was not a purebreed, but its health complications were unrelated to that fact; it is possible then that you could recover the difference in price between a purebreed and nonpurebreed, but not the full cost of the dog or the vet bills.