I'm looking for websites that offer movie transcripts for Japanese
films (in Japanese).  I'm a student of the Japanese language and fan
of Japanese movies and anime.  Being able to watch a Japanese movie
while having its subtitles or transcript handy would really be
helpful.

It doesn't answer your question, and of course the Japanese written
and spoken languages are two entirely different things to learn, but
you may want to check out Mangajin magazine. This was an American
magazine that reprinted Japanese manga but with a focus on learning
the Japanese language.

Mangajin
http://www.mangajin.com/mangajin/index.htm

Unfortunately it ceased publication a few years ago but there are many
back issues available online at the url above.

I searched the Japanese-language web for quite a while, and couldn't
find anything, other than links to scripts for English movies. It may
be that I'm using the wrong search terms, but I think that this sort
of thing may turn out to be hard to find.

However, I did find this list of DVDs that contain Japanese subtitles:
http://www.deaf.or.jp/video/dvd_new.html
And another list of only anime DVDs with Japanese subtitles:
http://members.at.infoseek.co.jp/yuzuusa/index-4.html

I have a few of these myself (Kurenai no Buta, Patlabor III) and they
are a lot of fun to study with, though of course you have to make
frequent use of the pause button! I hope these links help, and I'm
sorry I was unable to find any transcript files. Hopefully, another
researcher will have better luck than I did.