In which case, it may be that the transmission user does not have permissions to write to the filesystem (vfat has no inherent protection so by default it only allows writing by the root user).
You could try using the uid=n and gid=n options to specify the uid and gid of the transmission user, when mounting.
e.g.
mount -t vfat -o uid=1000,gid=1000 /dev/sda1 /mnt/USBstick
(assuming the tranmission uid and gid was 1000)