John Ball - the only zoo that @snowleopard notes in that post as holding Amur tigers and red pandas - is an AZA accredited zoo. Even so, there are no laws against other zoos obtaining them, AZA zoos just control most/nearly all of the population and don't share with non-approved facilities.
The vast majority (probably almost all) are bred in private facilities in the States, not imported from the wild. Non-accredited zoos will often breed their own animals and buy/sell/trade them with each other. A lot of primates in particular were also probably bred in medical labs before being sold off or donated when research ceased; a lot of the species you see in non-accredited zoos (olive baboon, vervet, rhesus and crab-eating macaque) were/are commonly used in medical research.
Most of these species are not endangered anyway, but it's worth noting that endangered species laws in our country vary widely depending on the species, origin, date of acquisition, purity, and which states or municipalities are involved.
