[Kransha]

Eagles, similar to Gandalf, are not people (or in this case birds) who intervene all the time. They are not legal citizens who would ally with man.

First, the only reasons the eagles came to Morannon in the first place was because Gwaihir, chief eagle, owed Gandalf a favor. They were not planning to help win the war and tried to stay as neutral as possible, even though they did hope the side of light would be victorious. The eagles wouldn't have been willing to play such a crucial role, since this was a war of man, not a war of Middle-Earth vs. Mordor. They had rights as sentient citizens of Arda and would not just take Frodo all the way.

Secondly, as someone else said, that wouldn't help all that much. Even though the journey would be easier, the job wouldn't be. If the eagles carried Frodo, man would not unite under Elessar. What about the great uruk-hai hosts? What about the evil men of the south and west? What about Sauron. If Frodo got to Mount Doom without affecting the rest of the world in any way, who cares if he gets the ring in or not? Sauron might be destroyed, but his armies would destroy everything with nothing to stand in their way. Plus, no last stand of men outside the Black Gate to distract Sauron while Frodo made his final decision. But, most importantly, NO SAM AND GOLLUM! If Frodo could just fly, no one would need to come with him except other eagles (even good warriors can't fight very well on birds' backs). Sam was Frodo's voice of reason, without which he would succumb before he reached Mordor, and Gollum was the device to destroy the ring, since his giddy dancing ended up doing the job just fine.