not by web ide, afaik.
if you can take snapshots, and (eg) it's on qcow2 storage, every snapshot is a raw file in the VM folder under <storage path>/images/<VMID>/
and has a similar naming schema as "base" disks,
if VM disks are named like vm-122-disk-1.qcow2
snapshots appear as vm-122-state-img_001.raw, vm-122-state-img_002.raw, etc.
Never tried, but, looking at my VMs storage folder, you should probably:
stop the VM to be cloned, clone it as usual on a storage that allows snapshots (like NFS fore above qcow2 example)
and then by CLI copy snapshots raw images in the destination folder, renaming them appropriately
note: the destination VM should have in its .conf a line like
vmstate: <storage name>:122/vm-122-state-img_002.raw
stating what snapshot file is in use now
and a line like
snaptime: 1438262544
stating when the snapshot in use was taken, I guess.
and a line like
parent: img_001
stating which disk image is the parent of the one in use, I guess
probably the best thing is copy the <VMID>.conf file from source to destination, adjusting manually VMIDs, MACS, storages, etc.
many guessing here, but hth
Marco