Here in germany, it is not customary to sell, say, live chickens or other live animals for immediate private consumption on a farmers' or weekly market. Since we can keep proper cooling chains, the one big advantage of selling live animals for direct consumption vanished decades ago.
- animals are sold (auctioned) at what would be literally specialized farmer's markets - but we call those fairs or auctions (or exhibitions) rather than markets.
- You can home slaughter for your own consupmtion, and of course you can buy the animals. That doesn't happen at the weekly grocery market, though.
In terms of zoonotic risks, we mitigate this by hygiene requirements.
I don't know whether there are specific regulations against selling live animals on farmers' markets (what I gather about the Chinese markets would anyways violate both general hygiene and animal welfare regulations).
But there are already economic reasons against that: buying live animals in order to slaughter and eat them immediately at home means a whole lot of work (and general mess), far more than buying some meat and cooking that. Plus the stress means a loss in meat quality: a farmer who cares about the quality of life for their animals would a) not want to expose them to such stress, and b) not be able to get the premium price for high quality meat. If you'd do the marketing according to animal welfare, it would be both more expensive and more work. The vast majority of people who don't have the practice, tools and space at their disposal could probably earn the money to buy organic meat in the time they need to slaughter and butcher the animal.
The big advantage of selling live animals for home consumption is that this is a way to keep the meat from spoiling without a proper cooling chain. Even including the hygienic problems of selling live animals for home slaughtering, it may still be the more hygienic option in certain parts of the world.
is allowed in Germany for home consumption (i.e. not for sale). You need to obey hygienic and animal welfare regulatiuns and the actual slaughtering needs to be done by someone who is sachkundig* (expert). Hygienic rules include animal health here (e.g. cattle needs to be seen by a vet before slaughtering). It is also possible and customary that you get an expert to do the actual killing, e.g. a butcher or a qualified hunter who shoots your cattle (or deer, see also below) on the pasture. Home slaughtering is done, but far more rare than, say, 40 years ago.
However, the customary way to do this is not buying the animal for immediate slaughter but to buy a young animal (or breed it), raise it and slaughter when it is "ripe". And someone who is doing that for personal consumption would typically not buy the animals at a market or shop or trader but at the local breeder's association or via their (online) classified ads (or those of the farmers' newspaper or even the local newspaper).
The alternative here to home slaughtering is to bring the animal to a butcher who can do the slaughtering and butchering far more easily where they have all the equipment - and then get back the meat and sausages. A similar procedure, btw., is customary for hunters: they kill and dress and if they don't have the equipment (e.g. a cooler that is large enough to hang a whole sow), a butcher either buys the dressed animal or processes it for the hunter. One difference is that the hunter is allowed to sell the venison (the hunter's exam includes relevant animal health and food hygiene for wildlife) whereas home slaughtered animals are not for sale.
One exception are carps. The traditional christmas/new year's eve carp was/is sold alive and then "watered" (kept in fresh water) for a week or so to remove any muddy taste from the lake it grew up in. Nowadays, also the vast majority of carps a sold dead (possibly chosen alive and then killed by the fish monger). Killing a carp again requires Sachkunde* since it is a vertebrate.
Traditionally, farmers' markets included livestock markets (just as there was a corn market and so on). I.e., farmers' market could be B2B in addition (cow is sold by farmer to butcher or to cattle trader) to the B2C meaning we nowadays associate with the term.
The bigger ones would rather be called a fair. Nowadays, many are fun fairs with only the name hinting at the live market, e.g. a local annual fair still had sheep and horse exhibitions by local breeders' associations till their 695th edition 2015. Apparenty, the Gallimarkt still has a livestock market section after more than 500 years.
OTOH, these fairs are more dry market than wet market (other than fun fair type fast food).
Instead of such livestock markets, AFAIK animal for slaughtering are nowadays sold either directly by the farmer to the butcher/slaughter house or to a livestock trader (livestock trader and animal transport is often one business).
"Livestock markets" or fairs still exist for stud (breeding) animals: these are fairs where the animals are exhibited, sold or auctioned but again they have nothing to do with grocery shopping.
For wild wild animals, catching them and bringing them to a market would already run foul of very general paragraphs of animal wellfare laws due to stress and animal that is not used to be so close to lots of humans will have.
In a sense the distinction wild animal vs. domesticated animal is a bit blurry. E.g. you can keep deer or mufflon (wild sheep) or even wild pigs. The difference to their wild cousins is that they are kept in a preserve, cannot mix with the truly wild ones and they are looked after by a vet if necessary. But like their wild cousins they are shot on the pasture.
Also in some neighboring regions it is customary to let domesticated animals range pretty much like wild ones, e.g. pigs in the Mediterranean.
Last but not least, already roaming herds of sheep or cattle grazing on pasture do have increased risk of certain zoonotic diseases. E.g. tuberculosis is very rare nowadays in Germany. One of the possibilities how you can nevertheless catch it is via drinking raw milk of a cow that in turn got boving tuberculosis by meeting a coughing deer on the pasture.
* Sachkunde in German means you have the required expertise to do something. I.e. a hardskill that comprises both theoretic knowledge and being able to do it properly in practice.