They would have been too difficult to hunt.
In, The Hobbit, Tolkien says:
The goblins hated the eagles and feared them, but could not reach
their lofty seats, or drive them from the mountains.
So, Sauron wouldn't have known exactly where the eagles were, and his orcs couldn't climb that high. If he would have sent out huge numbers of orcs that were trained in mountaineering, that would be a costly diversion of resources, with a low chance of success, since the eagles could always relocate.
Middle-Earth is about the size of Europe 1, and the eagles are extremely mobile. So if they fled their main nests in the northern Misty Mountains, it could take years for Sauron's forces to find their new home, at which point they could resettle again.
One might argue that Sauron could have hunted them with the Nazgul and fell beasts. The problem with that is that he didn't start using the fell beasts until a relatively late time. He probably had a limited number of them and wanted to save them for the campaigns against Rohan and Gondor.
Secondly, we don't know how fast or high fell beasts can fly. If they managed to fly to the highest peaks of the Misty Mountains and find the eagles, the eagles still could fly away or possible kill the fell beasts. Sauron wouldn't have wanted to risk that.