For full Hesperornis taming details, get the taming calculator app.
To tame a Hesperornis, you must kill a fish, and drag the fish's corpse to the Hesperornis to feed it. The fish's size plays a major factor in how many fish will be required (a 1.0x fish appears to fill up the taming bar twice as much as a 0.5x fish). Level of fish does not appear to impact taming. The Sabertooth Salmon is by far the most effective fish, equivalent of about 4 Coels of the same size, while the Piranha is equivalent to about 2.4 Coels.
Tame to this kill fish and drag there body to it and once tamed if you feed them more dead fish corpses it will build up and lay a golden egg which is used as a buff to certain dinos or oil or a player buff
Use a horse to lasso it, then you can move it around to new fish spawns when the area gets cleared. The horse can also step on the duck to pin it down, or simply use the lasso to keep it close while tribemates gather fish.
Can't be picked up by anything while they are wild. They do aggro on you, and can walk through door frames, so you can kite them to a taming pen then use an itchy to bring those succulent salmon to you. Tamed a 140 last night in about 20 minutes.