Hi Paula, not sure if I can answer your question entirely, but maybe I can share enough info and ideas, and others can chime in and fill in the blanks.
My over-the-door shoe organizer hangs by hooks. Like these:
first pic (this is the kind I have) &
second pic and the
shower organizer hangs by a rod with hook ends.
You have to put the hooks in to hang it, so instead of using the over-the-door hooks that are supplied, you might be able to use
these suction hooks or
these 3M Command hooks with
these strips. We use the suction hooks in the shower with the shower organizer.
I don't know if these ideas would work with the surface of the doors you speak of, or the way the doors travel. If the surface is too rough, suction definitely won't work. If the doors are pocket doors (which I doubt they are, but I am unfamiliar with them), then a door organizer of any kind won't work.
If the hooks will work, you could hang the shoe organizer as high as you can reach and cut the excess off of the bottom.
Hopefully this post makes sense (it's early

).