Since the lease agreement is concluded between you and the landlord, only the landlord is the competent person who can give you an invoice (and only if the apartment is rented for profit). Flatio can then only give you a digitally signed proof of payment acceptance.
TIP: Do you need to handle billing for your accounting? There is nothing easier than register to the Flatio for Business service, which deals with accommodation for employees. Learn more about Flatio for Business here.