As Huidong said, you will need root privileges to install Shiny Server.
That being said, there are hosting providers that can provide you with such an environment (often called a Virtual Private Server) -- on these you have root privileges and would be able to install Shiny Server on your machine. But in a shared hosting environment like the one you describe, you would not be able to set it up.
Some of the names to look at if you want to get into your own virtual server:
- Amazon AWS - The most popular choice right now, but they can be a little pricey.
- DigitalOcean - They're relatively new, but have a much lower entry point
- Dreamhost, GoDaddy, etc. (maybe even Bluehost, I'm unfamiliar with them) - In addition to the default "shared" hosting environments, many of these companies offer Virtual Private Servers for an additional fee on which you'd have root privileges.
One final option to keep in mind if you'd rather not manage your own server is our hosted service, http://shinyapps.io/. This service will manage the server for you and allow you to focus just on developing your Shiny Applications. It's currently in Alpha testing and is free for the time being.
Jeff