They did. And there was. Soylent Green.
Whoosh.
Suspected nothin’. They admit it. (Officially, it’s with signed consent, and they don’t sell the organs.)
Y’know, back to the topic at hand, would you believe I gave a speech on the subject last year?
Blew THAT class out of the water, I can tell ya. 
First you need to find a humane execution method that is going to give you a brain-dead, beating-heart donor with viable organs…
Make 'em watch TV.
But that’s not humane!
I don’t see why you couldn’t harvest corneas and skin and tendons and that sort of thing. Or bone marrow (which you wouldn’t even have to be dead for). If you’re going to kill a guy, why would there be any moral grounds to keep you from harvesting organs?
The argument I hear is that the practice would create an additional, commercial, incentive to condemn criminals to death. This could seriously affect sentencing. A judge who sees no value in a convicted man might send him to the chop block instead of giving him a life sentence since the convicts death could “do some good” for a transplant recipient.
I like the idea of putting a criminals’ organs to work after he’s done with them. At least then some benifit to society can be created.
BUT…
There is so much that could go wrong with the practice that I have some severe reservations about it.
I don’t see why they couldn’t do it the way the rest of us do it. Everyone I know has an organ donor sticker on their license. You just let the prison know that you’re a donor and they send your body to the harvestors when you die. No rush to off anyone.
I think there’s some confusion about a major point of the OP, and that’s probably my fault.
The default on this would still be kill 'em and bury 'em. If a condemned inmate made no choice, there would be no harvesting of organs. Inmates would have to volunteer AFTER the appeals process was finished; there would be no societal incentive to condemn more people to death.
I could see a lot of prisoners refusing to donate as a little “Fuck You” to the State. But still, it might get a few organs into new people if they asked the condemned to donate. They should wait to ask until after sentencing to help maintain the neutrality of the judge on the subject, if that’s possible.