Do I need OS for ARM Cortex-M0(3)?
You ask two very different questions.


  Will I be able to use operation system on ARM?


Of course, but it will likely be an RTOS like FreeRTOS rather than a dekstop OS like Linux.


  So do I need an OS in my case?


Strictly speaking: of course not. Any application can be written in a self-contained way, without an OS or RTOS.

But the question you should has asked is


  will I benefit from using an RTOS.


That is more difficult to answer. The Benefits of an RTOS are multithreading and the (additional) libraries, including hardware abstraction. The downside is that it takes time to learn how to use these facilities.

My gut feeling is that for the application you describe using an RTOS will not help you much, so the learning will cost you more than it saves you. But knowing an RTOS will probably be useful for some future project.