The best way for me to tame a Tapejara is to have lots of rare flowers, eat them, get on a flying dino (any) and get it in a taming pen, or if you don't have the resources to build one, have lots of bolas and keep throwing them at it, when the rare flower effect runs out, eat another one until it's knocked out, also, note that eating a rare flower will make the Tapejara agro on you, but not just you, also most wild life around you.
Use a fast creature - wolf, bear, raptor, sabertooth tiger - from whose saddle you can shoot, take a bola, and when tapejara lands, run there very fast! Tapejara will not have time to react and fly away, if you will throw the bola on the way to it. After this, knock it out and feed. It works on mobile, but I think, on PC it will works too.
ARRRRRRRGH I just had THE WORST taming experience! Pro tip: DON'T try taming a high level one by dangling from an argy by a grappling hook... You'll end up dying TWICE and losing your argy, guns, and 80 narcotics... >:( :(
Extremely difficult to tame. Best options are to bola it, and then use a longneck rifle with tranq darts. Otherwise trying to get it to fly into the side of a cliff and get stuff is the other option for a solo tame.
Since the new update, rare flowers make dinos aggro on to you, you can use this on a tapejara and it will follow you to a trap you've built. Just remember that each flower gives only 10 seconds of that debuff so if the bird is far away from your trap, you have to use it frequently while coming back. Also, if you need it to follow you after you've gotten off your dino, you need to eat another rare flower so that it can agro on to you once again.