What would an organ market need?
1. People who want organs would need to be able to find people willing to sell their organs. A kidney isn't like an unwanted Kegerator; it's not something you can post on craigslist. An organ needs to be compatible with the receiver. It seems crass, but that's why we have car dealerships - to match consumers with products. I'd envision something like a private adoption or surrogacy agency.
2. Parties need to be able to negotiate with minimal transactions costs. If Max decides he can part with a kidney, and Charles wants said kidney, the two need to be able to set a price that gives them both gains from the exchange. Again, an agency could probably help with this part - provided they could gain something as well.
3. Surgeons have to be included. There will be some doctors who will simply refuse to do these operations, just like there are doctors who don't perform abortions. The ones that are willing could attach themselves to matching agencies or set up private practices.
4. There needs to be a distinction between living organ sales and cadaveric organs. There are different implications for a living donor than for a deceased one. A living donor needs to factor in recovery and time-loss costs; obviously a dead donor doesn't. However, organs have to be transported and transplanted quickly after death. Again, I'm thinking of an agency-based system. You could agree to give organs a, b, and c to the company after death - if your organs can be successfully transplanted, then your family (or your cat, or a charity) would get $x compensation.
5. There has to be some kind of oversight. I know the economists who espouse organ markets cringe at the whisper of government intervention in the marketplace. However, some kind of government accreditation could help - buyers and sellers might feel safer knowing that their organ-matching agency met certain rules and regulations. In general, property laws need to be respected and enforced.
These are things that would help an organ market function in an ideal world in which all parties would be able to negotiate on equal footing. I realize we don't live in said world, and that if an organ market were implemented serious complications would arise.