This came up in the guest post by M. Hamilton, and I thought it was an interesting enough topic to be given its own post. I went through the covers of Justice League of America to see who did (and who didn't) appear on each cover through 1969. I broke it down 20 issues at a time:
As you can see, Superman and Batman only made rare appearances on the cover of JLA during the first 20 issues. Note that Green Arrow was not inducted until #4 (the Atom in #14, and Hawkman in #31), so aside from them, the Martian Manhunter was the first member not to appear on a cover, Green Arrow was the second, MM the third and then Aquaman became the first to miss two covers in a row.
In JLA #21-40 we see Superman and Batman becoming more regular features on the cover. Wonder Woman's consecutive string of covers ends with #25, while Green Lantern is first missing from #32. The Flash makes it to #33, but just barely, as his run of appearances includes this one from JLA #21:
Correction: As pointed out by sharp-eyed Tom Brevoort in the comments, those aren't the Flash's hands between Green Arrow and Superman, but actually Green Arrow's hands, rather awkwardly drawn (especially the right hand). So in fact Green Lantern had the longest run of the core JLA five. Good catch, Tom, you should be an editor. ;)
The Atom goes missing for a total of 19 issues, from 26-44; obviously his small size made including him on the cover somewhat difficult.
You can see Batmania writ large in this set of issues, as he appears on the cover for 13 consecutive issues, and the first one that doesn't include him has the adult Robin (of Earth 2) featured prominently. Batman was also the only JLA member to appear on the cover of the first issue of a JLA/JSA teamup between JLA #37 and #73, and the first JLA member to be the only JLA member on a cover with JLA #46. Of course that doesn't begin to tell the story as Batman was almost always in the foreground of these covers to take advantage of his newfound popularity with the TV show.
Doing rather well during this entire period was Green Arrow, who made the cover of JLA 40 times out of 74 issues. Not bad considering that his backup feature in World's Finest had been canceled in 1964 and that he seldom appeared anywhere else in the DC universe from 1964-1969. Aquaman, who had his own book, missed 43 of 77 covers, and only appeared on 11 covers after #28. Atom, also a headliner, only appeared on 8 covers from #21-#60.
Who made the most covers? The Flash appeared on 59 covers of JLA by the end of 1969, while Green Lantern made 58.