Yes! Scaling to many users is largely a function of computing resources, the type of Shiny apps you build, and whether you are running Shiny Server or Shiny Server Pro.
With Shiny Server Pro, you can specify the number of R processes any given application can run and how many connections can be mapped to those processes. Shiny Server Pro server will also handle turning people away, gracefully, if they are over the threshold; and lets you see how your Shiny apps are running and what resources are being consumed so you can tune your environment as the number of users grows.
You can download a 45 day evaluation of Shiny Server Pro and compare how your application behaves under load in both the open source and professional versions. If Shiny is the right platform and you need more than Shiny Server the open source version, Shiny Server Pro or hosted shinyapps.io may be the right answer.