Wyverns in a lot of ways are living weapons. They're huge (roughly the size of quetzalcoatlus), can fly, can support the weight of ONE unarmored human, and can breathe fire. Given these traits, it's likely that many that societies would try to capture and tame them for warfare!
The question I propose is, what would be the best way to capture a Wyvern with medieval European technology?
NOTE: magic does not exist in my story
DONT
steal its eggs and raise them as their own. Or kill the mother first then steal the eggs. You said it yourself they would be great for warfare and once captured depending on if its intelligent it wont do a thing that these tiny humans that it could easily eat says.A Wyvern would take many people to take down under the best circumstances.
Trap it You do not want to be fighting this thing head to head! FIRST find its cave get big heavy rocks and block it off then wait while it starves . Such a scary creature would need a lot to eat and wouldnt take to long for it to starve. I also bet a starving Wyvern couldn't breath fire as well that a healthy one can. After a few days or however long . You attack it a bind its wings and feet. Then feed it and hope it likes you.
The old fashion way
Find a injured wyvern and attempt to help it..... and hope it doesn't kill you. If it isn't injured leave out a bunch of meat and hope the wyvern comes. Do it again with it knowing you are there. Every time get a little closer make it know that you will always be there when it eats. Then one day you dont put out food and you are not there. After that continue doing it. Best case scenario the Wyvern associates you with food and doesnt kill you.
Same as anything. Ropes and nets.
Rope and Nets will wrap around their targets, weighing them down and tangling them. They can be safely launched or flung through the air, and don't need to contain sharp pointy parts to latch on. Its just a ton of fibers woven together. Historically, you can look at the Bola, Gladiator nets or thrown fishing nets.
Now you might think to yourself, how is a rope going to ever be heavy enough to weigh down something that can lift an entire person? Well just look at the medieval nautical industry. Ropes large enough to tie entire ships down to a dock, support a heavy anchor, and generally hung all over the place. The ropes can range from as thick as your thigh down to a single strand, so strength, length and availability are not really a huge issue.
As a bonus, you can weigh down the ropes and nets using small weights, and given a proper throwing or launching technique, it will be a simple matter to launch a net that fully expand, hit your Wyvern and then wrap around it tangling it, disabling its ability to fly and give you more time to wrap it in additional netting.
Example Scenario:
There's not just a problem with capturing the wyverns, but also being able to "break" (AKA domesticate) them.
For example, if you capture an adult wild horse, it is incredibly hard (some say it's just impossible) to break them. You need to have a horse from a young age--not necessarily birth, but pretty young, so they bond to you. Normally they start training them at 18-24 months.
For wyverns, trying to take a youngling from its mother would be dangerous. Therefore probably the easiest way to do this would be to steal eggs while the mother is away.
Note, that's just how you would get the first few. After a while, when you have a population of domesticated wyverns, it would be much easier to work with wyvern mothers that are domesticated themselves.
In this society, they would probably highly prize the people who have a good animal instinct for bonding with the wyverns.
Note that not all animals are tame-able. For example the zebra, contrary to appearing to be very similar to a horse, is not as related as you might think. People have tried taming zebras many times and failed. So with that rationale, you do have the option of claiming that your wyverns are the same way; their disposition isn't right for taming. Most predators are difficult to tame.
