Hi Wade,
Jamie here from Shopify. Neat idea! Gives me a fax machine vibe.
I imagine there are a few ways this can be accomplished. One that comes to mind would be to use our order notifications to our advantage. In your admin > settings > notifications > order notifications you can add a new email address. This could be somethign like neworders@example.com. We could then use somethign like this walkthrough (might work on Mac) on how to use outlook to automatically print incoming emails. I was also able to find this app (PC) that offers the ability to automatically print incoming emails. You can then customize your order notification template to include any relevant information for your warehouse team.
There are some tools built into Shopify that could help you manage your incoming orders when you are not in the office. You can add an order notification to *any* email, which could include some for of staff email that your team can check when you are not in the office. You mentioned you didn't want to give your staff access to your account, but I felt I should mention that you can create staff accounts and limit their access to different sections of your admin. For example they could log in and only see your orders, and not your theme, settings etc. This Shopify Doc goes into detail about this process.
I hope that gets you headed in the right direction! If anyone else has any other solutions I'd love to hear them as well!