For this use case I think you would want a virtual server on a cloud computing service (like Amazon Web Services, Google Cloud Computing, Digital Ocean, Azure, etc) with R installed and make use of the plumber package.
OK, that doesn't seem so complicated after all.
You say I can install the open-source R into a Linux server and build an API with Plumber package?
Should be worth investigating the documentation.
The development of the API would be easier on the RStudio environment but for deployment, you just need R, the plumber package and whatever package your API requires.