Yes that is possible. Just put old HDD in the case and then it becomes a usb External storage Harddrive when you plug it in. Windows will assign it a new Drive Letter pertinent to the usb port you use (the HDD will no longer be your Vista C: bootable Op Sys. It will be a mass storage device and you will be able to browse to your Documents Data to do as you want with)
Re-reading brambles post "Would this be possible" and my reply of "Yes that is possible" I feel I may have misunderstood what was referred to, so if you meant boot into the old Vista on HDD then my answer should have been: No, not possible to boot into Vista via this method (as per Fruit Bat /\0/\ already said).