This solution is not web-based!
There is no super friendly, web-based file manager for OMV. Feel free to create one.
There are plenty of cross platform options. While not super simple, jollyrogr's solution is cross platform. You could connect to shares via webdav. samba works very well and if you have remote users, just have them use vpn.
Some times, things just aren't easy especially when they are free.