Well, I tweaked around my way and was able to download all of those animations. So here they are for grabs, in swf format :D
How to open them? There are special Flash file players, but you can open them with pretty much any Internet browser (if you have the Flash plugin installed, of course).
I tried to convert them to easier formats, but most swf to video free programs suck or put huge watermarks. I was able, however, to convert them to .exe so you wouldn't need to worry about software. Also, you can pause some of them (not all, though D:) which is awesome for screencapping. Of course, this alternative is larger in file size.
Also, I feel really shy to ask this, but I'd love to discuss the plot or the characters with people, but I'm only friends with a couple of persons into it. If I were to make, say, a weekly post for discussions about Okami, would you guys be interested in participating?