Most exhaustive source: vg_footage, an absolutely complete video playthrough of FFX.
The advantage is that it has everything. The disadvantage is that it's small size and compressed file format, so the majority of screencaps are a little too small and fuzzy to be usable. Also, one has to make one's own screencaps. I have made a little file of over 500 Auron/Lulu screencaps from it.
I use them sometimes (my "she hit me", "you're the one running" and "no matter how dark the night" icons, for example), but mostly I scour online archives for larger-sized screencaps.
Favorite screenshot archive:
FFWorld, because it has more large-size screencaps of game play (as opposed to FMV) than anyone else. Also it has FMV caps under "images", so it's one-stop-shopping.
Favorite FMV screencap archives:
And if I can't find what I need there, I poke around on somewhat hit-or-miss sites:
RPG Gamer (mostly gameplay)
FF Insider (Mostly gameplay, only goes thru Kilika, useful for Rikku & Al Bhed sequence)
Blue Laguna (Mostly FMV)
Moby Games (A few FMV)
Warcry (Utterly random)
Google image searches are also useful. Now and then I still find a good picture (e.g. "blitzball") that I don't on any of the bigger archives.