An inmates organs are just as good as the organs of a non-convict. If they want to make amends with whatever crime they did and they are able to donate an organ, they should be able to. This is a good way for inmates to participate in society in a healthy way.
Inmates should be able to donate healthy organs if they so choose. There is no reason that their organs should be passed over for prospective transplant matches simply because they were committed to prison. In imposing such limitations, it is the prospective transplant patients and not the inmates who ultimately suffer the most.
I believe inmates should be able to donate organs to the thousands of people that need them. If that is something they would like to offer to the world, then they should be allowed to do so. It shouldn't matter if they are an inmate and are healthy enough for the organs to be viable.
While it is a bit of a logistical nightmare, to me it's an easy decision as these organs would save many lives. There is a huge shortage of donors and while this won't solve that problem it would definitely be a step in the right direction. I also like this because it would give an inmate some chance of redemption as well.
Assuming that the question refers to inmates on death row (because I know of no prohibition for other inmates against organ donation), it is not feasible. The only approved method of execution is lethal injection (though the gas chamber is allowable as a second choice in some states). Lethal injection does irreparable harm to most of the major organs.