When you access a file from the Altavault, one of two things happens -- it either pulls from the local cache or pulls from the cloud (if the file has been evicted).
You can do this ad hoc (when you need the file), or you can use the "prepopulate" feature to mass pull down a whole folder. In either case, the files are then ready for local restore on your LAN.
The details of this are a little different for clouds like AWS Glacier. Since the restore times can be like 4 hours, it's best to use the prepopulate feature so you can track what is coming down. (See FAQ for more details https://aws.amazon.com/glacier/faqs/)
Does this help?