With Shiny, can my customers analyze data interactively via the web?

Yes. You can see examples of Shiny apps similar to ones you  might create for your customers at shiny.rstudio.com.

To share your Shiny apps with customers online you will need a Shiny Server. The basic Shiny Server works for many needs. To scale to a large number of users or processes you should consider Shiny Server Pro or ShinyApps.io.