Typically, an institutional IT department would assist you in setting up a server using whatever processes they use for Linux web server management.
I manage the Runestone WeBWorK hosting service. If it is helpful, you can use this service while keeping the students completely anonymous. You would have to forego some features (notably a direct connection to your LMS and email features) and also you would have to tolerate some small tedium as the instructor managing a mapping of anonymous user IDs to the actual IDs of your students. If you would like to pursue that, I can go into more detail.