Not every company-internal communication needs to take the form of a letter.
We do almost everything in my company by e-mail, especially since the employees are scattered around a multitude of cities in Germany as well as multiple countries and continents.
When necessary, people write what might count as a brief letter (such as messages from HR). However, most of the time, the e-mails are just assignments of projects, status updates, questions, or informing project managers that a translation is ready for delivery. It is an unwritten rule that such communications should be be kept as brief as possible (unnecessary wordiness is a waste of time).
After I have completed a certain type of job for certain customers, another department needs to spend a minute or two to create another type of deliverable from what I delivered. Either the project manager could write a separate e-mail to inform them that the job is ready for them to do their magic, or I could let them know.
Since I have to let the PM know by e-mail that I've finished anyway (which usually consists of a very brief greeting, then "done" followed by a path to the deliverable. The PMs typically respond with "Vielen Fank, hb." followed by their automatic signature), when the other department needs to do their bit, I simply include them in the CC:, and after my brief message to the PM, write "@[Department name]: please create the....." Often, the PMs will do the same with me when my work follows that of those to whom the project is first assigned. It's simply easier to identify.
We're all happy with that and it works quite well. No one considers it impolite, offensive, or out of place. And, given the number of e-mails that fly back and forth, it saves a bunch of time for everyone.