There is nothing built in to Store, nor is there an extension that would allow something like this.
While being a bit more work for you, you could use something like PDF Press and the {exp:store:orders}
tags inside of a template that requires you to be an admin to access. With something like this you could view hundreds of orders at a time.
Alternatively you could use Store's email templates and an email rule for orders. Basically you would set up an email template in Store for admins, then send it to a specific email address, then the client could easily set up an email rule on their computer to print any email with a specific subject. I have a client that has this exact setup on a site and it has worked out well for them.
Finally your other option would be to build an extension to add the functionality to Store. You would probably be best to go with the store_order_complete_end hook if you wanted to do something automatically after each order. Or use the Store Reporting hook if you wanted to modify how you access reports in the CP.