So, we have the Quick Stock(QS) in shops that help you move items, but when you have your inventory you don't.

When you are in categories mode in your inventory you can find like items pretty quickly, but you have to then find each individual item in the QS screen to send them where you want.

I would like to see a Vault/Shop/etc. QS option in the inventory when sorting by Categories.

Example: I have a group of 30 food items that I want in my vault, but I have a total of 200 in my inventory. So instead of going through QS and tracking down each item by name, I can then just click a button on that category and send all of those items away.

Obviously more elegant than this,

I am not sure this would work because a user can have multiple shops. Not opposed to it though

I like the idea but I would suggest a different solution: make Quick Stock(QS) able to be sorted by category, should help a lot in making things easy to find in there.

this is better because the inventory page won't need to get a form added (that is how QS works) and that means the inventory page doesn't get complicated