I've been in a similar situation in a global organization. I had no quick fix.
I started by screen recording myself demoing a Shiny app and some Rmarkdown reports on a trial installation of Connect on a laptop. I posted that video to our internal Workplace app as well as Confluence page, and that got a few people's attention. From there, the video demo was shared, and I expanded my network of interested people until a particular director - who had time, money, and people - was interested.
After several pitches and slide decks outlining and justifying requirements, that director assigned a software engineer and an IT person to get Connect installed and running on the correct server on the correct network. I connected the software engineer and IT person with my RStudio Customer Success rep so they could work details among themselves. The software engineer would occasionally ask me to check a URL to see if Connect was up and running yet. The IT security person had a few questions along the way, and I would direct him to the Connect Admin Guide here https://docs.rstudio.com/connect/admin/ as well as directly to my RStudio rep or RStudio support.