Have you heard of the htmlwidgets
package? It basically allows you to integrate your custom Javascript widgets into Shiny (or simply call them from R as you would with a standard plot function) by letting you build an R binding function to the Javascript code.
Here’s a good tutorial to learn how to do this: http://shiny.rstudio.com/tutorial/js-lesson1/. It might look a bit daunting initially, specially if you are new to Shiny, but if you follow through the tutorial carefully it would only take you a day or two to build your first widget.
Keep in mind that you need to build a package (or packages) to host your widgets before you can deploy in Shiny, so make sure your apps have access to those in the R environment they will be running in or else they won’t render and your app might even crash at runtime.