When you first encounter a Tapejara in Ark Survival Evolved, you realize that these flying reptiles are the most skittish creatures in the game. Luckily Tapejara can be ensnared by bolas, the problem is getting close enough to one. Sometimes you can get lucky and surprise one, or scare it into the side of a cliff where it will get stuck, but the following method works much better.
One cool mechanic in Ark is when you eat a rare flower. Eating a rare flower near almost any creature will cause them to aggro on you. This even works with completely passive creatures like dodos, Pteranodons, parasaurs etc. Put some rare flowers in your hot bar, get fairly close to a Tapejar, then eat a rare flower. The Tapejare will turn on you an attack for a short duration of time. When it does that you can hit it with a bola and start knocking it out with crossbow, longneck rifle, or whatever you have. A harpoon launcher with a net projectile will also work, but I like the bola because there can be a lot of other creatures that attack you at the same time, and many can be quickly with a bola, which are cheap and faster to use then the net. Before all of this check for dinos that you are not going to be able to handle, like brontos, for example, because they are going to end your Tapejara taming for sure. The bola trap is only going to hold for a minute, and there is a cool-down before you can use another one on the same creature.
If you find a high-level Tapejar you may want to lure it to a trap, this is probably going to take a number of rare flowers, and you are bound to get hurt doing it. I just tame them out in the open, you risk a lot that way too. Good luck taming a Tapejar. They are really cool creatures, and are probably the best for flying around with a friend.