I don't think your shinyapp can help the user download directly from ftp to the computer. I think one solution is to have your shiny apps to download the files first from the ftp to a tmp folder, then your user downloads the files from the shinyapp to his computer. And this is completely feasible.
however, but I don't have ftp to try, it is possible that if you provide to the client in browser a link to ftp, it would be like http and the browser can download directly. But I am not sure.
Otherwise, you could see your shiny apps as a helper to know which command to use. You shinyapps could show the user the code that needs to be run locally.
Just some ideas...