Ok, transplants for a body that is brain-dead. Buddhism says no, don't do it. Brain-dead doesn't mean you are dead. There are real life stories where people have been in comas for years and years, and occasionally they come out. Where were they? They don't even know, but they were still somewhere! Rosemary's grandfather, in particular, was a very active man, a millionaire, extremely active. When he was about 80-years-old, he had a stroke and he basically became a vegetable, he sat in a chair. If you put candies in front of him he would gobble them down, but otherwise he would do nothing, he would stare blankly at everyone. Eight or nine years go by, and he just sits in his chair. His wife was a similar age. When they were a bit over ninety, his wife died of old age. Two or three days later he became aware. He had brilliant conversations with Rosemary's mother, and some of her siblings, and then a couple of days later he died. Where was he? He didn't know. That's one stroke situation. There are some people who are declared hopeless cases, deep comas and such, and every once in a while they come out of it.
When doctors use the term "brain-dead", to Buddhism, that is not actually dead. So transplants are taking part of the body out when a person is not actually dead. Maybe they are going to die, but they are not dead yet. By taking a part of the body out, Buddhism says you are disturbing that person who is basically still there somehow.
Now, to go into more of the philosophy and theory of Buddhism. And I'm not saying this from experience. In many cases like this, what we call in English a spirit, a mind, a soul, whatever, is not quite the same thing as in Buddhism. Buddhism uses the word Vinnana, which is often translated as "consciousness." Vinnana will get out of a dying body but sometimes wait and hang around the body. It may still be attached to the body and it can't get a proper rebirth yet. It can't get a rebirth yet because it is still attached to the body and/or attached to the people around the body, their loved ones, husbands, wives, children, whomever. This stops it from just leaving and getting reborn. Disturbing the body at this time can affect it in such a negative way that it will get angry, it will get upset, all sorts of negative things can happen for the person dying which may affect their rebirth in a negative way.
So basically if we consider Buddhist philosophy, we would not want to create difficulties for the consciousness in the critical period of transition to a next life so Buddhism, would be against transplants, taking parts of a body out when it is only brain-dead. Once the body is truly dead, that can be a different story.