i assume the new volume is already formatted..
mount it to eg /mnt/storage
rsync -a /var/www/ /mnt/storage/
do this a couple of times until the sync time is as short as possible.
set the new volume to be mounted at /var/www in /etc/fstab, but do not actually mount it there yet.
stop the webserver service, eg systemctl stop apache2
stop any other services that might write to /var/www or might restart the webserver service.
do a final resync,
unmount /mnt/storage and remount it to /var/www and restart the stopped services.
(or just reboot)
at some point, once you're confident this all worked without problems, you will probably want to stop any service that writes to /var/www again, unmount the new volume, then delete everything in the original /var/www on the root partition to free up that disk space, then remount the new volume and restart the stopped services