Hi, as far as I know, there're 2 possible reasons. 1: Your android system need to be updated to newer version, some old versions (quite old, like 2.1) don't accept app moving. 2: Many apps can't be moved to SD card, especially pre-installed apps. Try to clean your storage, removing temporary files and unnecessary apps. Hope it helps.
There's another way to do it without rooting:
It won't move everything but will allow you to move some where the option was greyed out. I've used this for both of my daughters phones and it works fine.