How to approach universities to sell them software licenses for all their students

Based on my experience, it is very hard to sell for the whole uni in one step, especially for large institutions. I would approach first a single department like Computer Science, Economics, or Literature and one specific professor with a large relevant course there. Give them a free trial first and convince them to integrate your service into their curriculum. It is like a pilot project in an enterprise sense. They might become your "internal champions," and you can go one level up in the org and convince the head of the department and, eventually, the whole uni. It might take 3-5 years to go through the entire process because of the nature of the university course development lifecycle.

This will be challenging unless you offer it free. Universities are very bureaucratic and things move slow.

I teach in the US and I can choose appropriate software if it works for the material I am teaching and it is free or already available through the school IT dept. For example I have encouraged students to use free version of Github, Geany IDE, Pycharm, git, Sourcetree App, and Visual Studio. Why? I can tell you 4 of these were because I use them personally, the other 2 are what the other teachers are using.

But when you want the users to pay - that's another topic. I've been told by fellow professors I need to talk to my "boss" (the dean) to get anything paid approved. 1 year in, I have never tried. The barrier is very high.

Many times software companies offer free licenses to students and academic users (for example Jetbrains does this) because -- I assume this is the reason -- if they can hook the users in early, those users will stay loyal and pay after they graduate and can afford to pay.

Whatever you choose to do -- do not crawl for email addresses and send spammy messages to professors. Those messages, if they do come through, come off as extremely sketchy.

BTW, have you tried Overleaf? Your app reminds me of theirs except overleaf is all LaTeX.

Reviving this old thread, as I have similar questions, and I'm finding answers one email rejection at a time! :) Out of curiosity, @WolfOliver can you share where you are now with this project since your original post?
I agree with the points made by @szferi, @neea, @blakerson, @rywils21. I'll add a few more observations:

1. Whenever it comes to purchasing s/w, depending on the type of the s/w, there are different higher education (HE) groups with very different power structures: faculty, students, administration (could be separate departments as well as organization-wide), and some relevant staff groups (could be IT, or academic technology, or center of teaching and learning -- there is no end to it). Chances are that the people you know at different colleges/universities would belong to one of those groups -- talk to them to find out how they adopt new s/w.

2. If you are trying to reach those who write a lot, think of grad students (thesis), and postdocs. All big US universities have grad student assoc./postdoc assoc. See if you can get in touch with any of them. I'm assuming you already performed some sort of user research to understand what it'd take them to try a new writing s/w.

3. Acad. conferences could be a good place to meet a bunch of people who write a lot (and wish they could publish even more papers!). See if you can get in one as an attendee to talk to the other attendees. It could be cheaper to buy a year-long membership + the conference registration than to signup as a vendor. If you have money for the latter, power to you!

4. Given the abundance of FOSS in this space, and relative poverty of students/scholars, the value proposition of your offering has to stand out. Have you tried experimentation around how much interest you can generate if it's available for cheaper (or, even free)? Anything that tells us about the elasticity of demand ( say, by running a x% discount campaign for a week or something) is super valuable. Of course, you do need a reasonable traffic/download rate so you have statistically meaningful results.