We have existing website built in C# and hosted on a shared Windows server. The aim is to build a simple app within existing website that would take in some data, entered by user, and process it using some R scripts (some statistics) and publish the output on the website in a static (dynamic?) format.
A Shiny app can absolutely be integrated into a webpage, typically as an iframe hosted by shinyapps.io, and embedded into your website. See Shiny homepage for an example. You can see some pointers in Embedding a Shiny application in a html document on Google Groups.
A solution more in line with what was asked is to use a server application like RServe to facilitate communication between a running R session and a .NET/C# application. Since RServe allows for remote connections, it can be used by an existing website, with the RServer running on a different server.