In Deep Geek did a video about this. And there are some good reasons. Essentially the eagles wouldn’t want to because it’s not really their mission. Or to put it another way it’s not really what they are designed for. They are supposed to send messages and not carry the enormous burden that is the One Ring. It’s like trying to send a heavy rock somewhere with a paper airplane.

The quote from the book referenced in the video is:

“How far can you bear me?” I said to Gwaihir. “Many leagues,” said he, “But not to the ends of the Earth. I was sent to bear tidings not burdens.”

Also, a commenter made a good point that the eagles themselves would be susceptible to the Ring’s influence as well.

The Eagles were also described as the proudest of beasts. They would have been the most vulnerable to the Ring’s influence themselves.

https://www.youtube.com/watch?v=alOGSooSqmM

I remember seeing a clip of him in an interview, and he says that several times he’s been asked why they walked to Mordor rather than using the eagles to fly there: “and I’ll tell you what I tell them - shut up”