What makes you think that any business MUST accept it...you're correct that the vet would have to accept LESS $ (the interest is actually paid up front by the vet/vendor)...Care credit is not a credit card, the vet must accept it because you get 12-18 months interest free so how they make there money is the vet accepts less money from care credit and you still pay the same bill.
for the past 20+ years I have refused any type of charge cards in my business...why should I give 2-3% to the bank!