What is "move for"?

It means due to the presence. It's the same usage as the expression "Can't see the forest for the trees."

This usage is not current in my (American) dialect. If I was to use this, I would surely say "You can't move for all the books in her room," but I would be more likely to say something like, "You can't move, what with all the books in her room" or "You can't move because of the books in her room."

"Move for" has a different meaning, in parliamentary and legal procedure, relating to make a motion. "I move for Ms Lee's remarks to be stricken from the record." "I move for immediate dismissal of all the charges."