I'll stay clear of Cultured Bioware, and maybe even Bioware alltogether, as black market items, but I think from a RPG perspective black market ware is definitely a thing. You have organized crime harvesting cyberware from unfortunates and selling them off as used? Definitely. You have shitty back alley clinics implanting whatever you bring them? Absolutely. This is a dark and shitty world where your ideals of the perfect implant is just about as far away as your chance of becoming ARES CEO. You take what you can get, and a good GM always keeps his players hungry.
Oh, for sure there is a black market out here for body mods. Just not the organic kind. Cyberware, military skillsofts, weapons, heck even good ol' fashioned clothing are all staples of the Black Market. (You used to be able to make a pretty penny bringing new Levi jeans into Russia back before the fall of the wall... Buy them in the states for $60. Sell them in Russia for $300...)
Basically the Black market handles stuff that doesn't spoil, or is relatively easy to store.
"Squishy" things, like organs, end up being handled a different way as pointed out. Tanamous is the current booggieman for this type of shit with their "tag and track" inventory. (They keep a list of DNA and blood types to match up to prospective clients. A client comes to them and needs a new kidney, they look in their records, find the match... and go get it! Too bad for the living "Fridge" its being stored in... but hey, they still have one more kidney. For next time.)
There is lots for your character to work with.