Reposting a few links from my Twitter reply, plus a few other thoughts...
First, some resources from @nathan:
This PDF with some common questions and answers for IT is also a good resource for starting a conversation about supporting R.
If your organization is a Windows-only shop but you'd like to use RStudio Server, Shiny Server, or RStudio Connect, ask IT if they can provision a Virtual Machine with a supported distribution of Linux installed. If they tell you that they don't have any experience working with Linux, ask around! If you're part of a large organization, you might find that someone has had to set up and maintain a Linux server at some point. If all else fails and you're up to the challenge, try setting a server up yourself. You may find that it's not as difficult as it seems at first, and you'll build some valuable skills in the process.
I agree that demonstrating how powerful Shiny and R Markdown are can kickstart the process. A lot of decision-makers and other folks in IT don't understand how Shiny is different than other dashboarding tools you may already have that just filter/aggregate a table of data. If you can build a killer app to demonstrate that it's more powerful than whatever tools you already have access, it may help convince them to support the tools you need. (@nathan and his team have put together some great examples at solutions.rstudio.com.)
If you're considering one of RStudio's commercial products like RStudio Server Pro or RStudio Connect, you can always reach out to our sales team even if you're not sure if you have the infrastructure available to support those products. If you're hoping to start with an open source copy of RStudio Server or Shiny Server, this forum is a great place to get some advice.
A few comments have suggested that it's easier to acquire the tools that you need if you're able to and ask for forgiveness later. I think that's a fairly common way that these tools come to be supported, and something I've experienced in the past. But my personal opinion is that things will go much more smoothly long-term if you're able to have a conversation with IT first and get everyone to buy into the ecosystem.