Simple LMS for uploading assignments

If you are sure your functional requirements are not going to grow over time, then you could use a simple form like Gravity with a file upload option :
https://docs.gravityforms.com/file-upload/
https://websitetutorials.grafix.gr/wordpress-2/wordpress-plugins/gravity-forms/protecting-you-gravity-forms-upload-files/

It will timestamp the entry, and you can ask for any other data from the user.
I use this for resumes, and for bursary applicants, etc.

If you need something more complex, and can grow in function, use LifterLMS :
https://www.ads-software.com/plugins/lifterlms/

Thanks a lot! My school have used an expensive commersial LMS for years, but the only functions I am actually using is a blog, and an upload function for student assignments.

The blog function can for sure be done with WordPress and I guess the assignment upload as well.

Yes, for sure. It’s a simple method. I would actually recommend putting your efforts into protecting the files.
Building the form with file uploads capabilities is the easy part.
Here are some of the more popular approaches :
https://ninjaforms.com/
https://wpforms.com/
https://www.gravityforms.com/