I'm currently able to run pandoc on a web server myself.
I just made a Dockerfile that derives from the offical python dockerfile,
After that you can just run:
RUN pip install pypandoc
RUN python -c "import pypandoc; pypandoc.download_pandoc()"
At this point, you'd have a working "web server" with pandoc,
you'd just have to write some code to actually make calls to pandoc in any way you see fit.
You could use a web framework like flask and just wrap an api around it.
You can do a similar thing in almost any language of your choice by making system calls with the text you'd like to convert.