The question Did Voldemort seek out Phoenix tears? got me to wondering if there are magical-animal zoos (in England or Europe) where wizards can view creatures they might not otherwise have access to (like dragons, phoenixes, unicorns, etc). If so, they might have provided a more convenient method of access to a phoenix for Voldemort than Fawkes or Sparky.
I don't think it's a stretch to assume there certainly could be magical zoos. While it doesn't explicitly say so, there are reserves for different magical creatures.
Perhaps the most important step in the concealment of magical creatures is the creation of safe habitats. Muggle-Repelling Charms prevent trespassers into the forests where centaurs and unicorns live, and on the lakes and rivers set aside for the use of merpeople.
In extreme cases, such as that of the Quintaped, whole areas have been made unplottable. Some of these safe areas must be kept under constant wizarding supervision; for example, dragon reservations.
Fantastic Beasts and Where To Find Them - pages 9-10 - Scholastic
While their might be magic zoos, I can't see phoenixes in them. Not only would keeping them in a cage seem difficult considering their magical powers, it wouldn't be a very dignified way to keep them. Not that humans ever care, though.