Or instead of running away from copyright infringement, you could
dispute it using a real reason why you think your AMVs should be allowed. Fair Use. (there's usually a link for disputes, or follow the link Nya posted) Just say something like: "By combining the music with the rearranged clips in the manner shown, this video creates an entirely different experience than the original works used; thus falling under fair use in copyright law." Or something like that. I mean... you truly believe your AMVs should be allowed don't you? Just explain why. (which is kind-of a difficult task, but at least you can try...)
And it also depends how you do it. Acknowledge that you are using someone else' work by giving credit where credit is due (and that's just good form... otherwise it's almost plagiarism in a way... I mean you can write an entire academic paper using ideas from other peoples' work as long as you cite everything... and you could argue that AMVs are no different...) So provide information/links to the artist/anime in the description, and maybe even add a disclaimer like the one used on the org. THIS VIDEO IS PURELY FAN-MADE AND IS NOT ASSOCIATED WITH THE MUSICAL ARTIST OR ANIME COMPANY IN ANY WAY. It's just a simple way to let other people know that the work isn't fully yours.
Then again, even that sometimes doesn't work...
...but the way I figure... the more people dispute fair use, the more likely AMVs as a hobby will gain some legitimacy.
Otherwise the hobby is always gonna be kept down by The Man
Row Row, Fight da Powah!
/mini rant
Anyways, the org still makes more sense for collecting AMVs

Also, you might find this useful for what you are trying to do:
The Ultimate AMV ResourceBest of luck!