I believe it depends on the company policy - what and how they wish to track the process that and on country legislation - what they absolutely have to track and show in their books.
In some countries even if you give away free products you have to pay VAT, sometimes you have additional taxes, so you need to issue an official invoice with the correct number range and you show this in your books.
Some companies define/or should define each of the employees as separate customer numbers, in others they opt for one-time customer or dummy customer per department (especially if they use an external system or some special software for that process).
From what you have described, it seems that you need to perform requirements gathering, spend some time with people from HR, Finance, Logistics.
Why HR? Well, you may need to access sensitive data about employees to determine what quantity and when to give to each employee (it could be even in a separate system). You need to know what employees are eligible to get based on the group contract (I am not sure if this is the exact English term), what internal controls are established to avoid misuse, what documents and reports are needed for different departments.
Why Finance/Controlling? Because even if you don't profit from these books, you spent money to produce them, you also wish to track where your money went, you have to show this in your books. FI/CO would know what postings they expect to see. One more thing - in some countries zero value invoices are not allowed at all, so check with the your legal adviser how this process is treated.
Why Logistics? Because you do not know when the books are requested, where they are stored, when and how people pick them from the warehouse, how and when these books are shipped to other locations (imagine an office in the city center and a warehouse 30 km from there). Scheduling is quite important, also it is very important to know the implications on product availability - what if a customer wishes to buy the same book and you do not have sufficient stock to cover both his his requirement and employees' requirement?
Can employees choose which book they want? If yes - how, when? Can you simply go to the warehouse and take the books from there?
Who and how orders the books? Every employee fills a form by choosing products from a catalog in a company website, or writes mails, prints requests etc.? Or the team lead does that for all his subordinates? Does this depend on contract type - permanent, part-time, temporary? What are the reporting requirements, do you need special printforms? Do you need traceability per employee or per department/per location is sufficient for invoicing?
Seriously, you cannot prepare a meaningful functional specification if you skip the requirements gathering phase. Organize a workshop/conference call and discuss the open topics before you propose a solution to the client.
You do not need a 'special' program logic that will automatically create a '0' value invoice, this could probably be achieved with proper customizing and some job. If you need one invoice per location, then maybe creating 7 invoices manually via VF04 is not such a big deal.