Your question divides in two parts, on related to LaTeX, one to law:
1 How can I create an letterhead with LaTeX?
2 Am I allowed to do so?
To Question 2: If you are a member of a university (for example a professor) and have to write notes, letters in the name of the university then you should use the official letterhead. That makes your note or letter to an official document announcing an offical statement of the university or part of it. And has the advantage that it reproduces the corporate identity ...
An academic job applications is never an statement of the university, it is a personal thing for you. So you never should use a letterhead of your university (and as far as I know you are not allowed to). It is not the university which wants a new job, it's you. In your CV the reader will see that your last activity has been on the university.
See the comments of the other users: At last they tell you to avoid this!
Conclusion: No (as already mentioned in the linked discussion)!
To Question 1: A "good" university will provide templates for letters and other documents and if they knew and use LaTeX, they will have templates for LaTeX too. So first ask for the official template.
If they have no LaTeX template LaTeX gives you several possibilities to create a letter, for example the class
scrlttr2 (KOMA-Script). At last it depends on the corporate identity how the letterhead has to look.
One possibility is mentioned by the comment of Sveinung:
I think this is a very good point to start. I'm using KOMA-Script too :-)
The other possibility is to use
letter, but I have no example for that.
Conclusion: It depends on the corporate identity of the university how the letterhead has to look. Usually the university will provide a template. If not you can make it with LaTeX by your own, but it will be not so easy to keep the corporate identity.
Because you gave no example I can't say more.