I bet your app has install.packages() calls as part of its app code. Never do that; instead, install the necessary packages from the terminal, while logged in as either the user "shiny" or as root. From the terminal you can switch to the shiny user with the command sudo su - shiny (that's a dash with a space on BOTH sides), then run R and call install.packages() as normal.
Keep in mind installing or upgrading packages will, by default, affect all apps on a server. Use something like packrat if you want to isolate each app to have its own private library.