In the Monster Manual, it states that wyverns can be trained as mounts. Other than that, there is no other reference to it. So how does one accomplish this? I imagine it has to be animal handling checks, but I'm not certain.
No real details exist.
Mounts other than those listed here are available in the worlds of D&D, but they are rare and not normally available for purchase. These include flying mounts (pegasi, griffons, hippogriffs, and similar animals) and even aquatic mounts (giant sea horses, for example). Acquiring such a mount often means securing an egg and raising the creature yourself, making a bargain with a powerful entity, or negotiating with the mount itself.
This glosses over the idea that it is possible and some examples of how, but the DM makes the final call on this. Even in the Down Time additions in UA they don't include it that I saw but it could easily be fleshed out using the existing ones in the PHB and the UA as guidelines for costs and issues that could arise, such as when it tries to fly it spooks local livestock and you have to pay extra as recompense for the inconvenience.
A great many things are simply left to the DM and Table's decision and this would be one of them. Work with your DM or your players for a reasonable system or play it by ear as no two animals have the same personality.