Is it possible to mass move things to your vault and if so; how?

You need to create a shop or gallery first, and then you can either navigate through your shop/gallery to the quickstock feature or hover over the commerce tab in the upper right > Your shops > the (qs) link next to any of them to mass move items to your shop/gallery/vault/wardrobe/etc. :)

commerce > your shops > (qs)

Ooh! Thank you both! Now I can move things much faster when my inventory is full

Glad to help! I didn't see Darkrai's post, sorry about that