I've scoured every Tolkien book I have, and I can't find any indication of population size. The best I can do is "many", from a very early description of Eagles' Cleft, in the mountains encircling Gondolin in the First Age:
[T]he path is narrow, and of the right or westerly hand a sheer wall rises nigh seven chains from the way, ere it bursts atop into jagged pinnacles where are many eyries. There dwells Thorondor King of Eagles, Lord of the Thornhoth, whom the Eldar named Sorontur.
History of Middle-earth II The Book of Lost Tales Part Two Chapter 3: "The Fall of Gondolin"
This passage was written before 1920, to give you a sense of just how old it is; whether you consider it canon is really up to you.
We can, however, be fairly confident that the Eagles do reproduce; in Return of the King, Tolkien writes that Gawihir was a descendent of Thorondor:
There came Gwaihir the Windlord, and Landroval his brother, greatest of all the Eagles of the North, mightiest of the descendants of old Thorondor, who built his eyries in the inaccessible peaks of the Encircling Mountains when Middle-earth was young. Behind them in long swift lines came all their vassals from the northern mountains, speeding on a gathering wind. Straight down upon the Nazgûl they bore, stooping suddenly out of the high airs, and the rush of their wide wings as they passed over was like a gale.
Return of the King Book VI Chapter 4: "The Field of Cormallen"
That quote further suggests that there were a fair number of Eagles, though it's impossible to even guess at how many.