Use the gid and umask options to mount, e.g. if the group you want to have write access has numeric ID 500:
mount -o uid=0,gid=500,umask=002 <device> <mount point>
You can add this to the options field in /etc/fstab if you have this mount set up in there, so the options get applied whenever you mount it.
-- MSDOS based filesystem don't do *nix style permissions, so Linux fakes it for you.
-- You use the uid and gid options to determine owner and group of files on vfat volumes.
-- The umask gives what permissions are denied. A umask of 002 denies no permissions for user and group, and denies write permission for others.