Is there a way to share shiny apps internally with non-R users for free with unlimited hours? Also, due to the sensitivity of the data, I'd prefer not to have it open to the public.
Yes. The best solution is: https://www.rstudio.com/products/rstudio-server-pro/ but if you want a free version you can get: https://www.rstudio.com/products/shiny/shiny-server/
and for example add: https://www.shinyproxy.io/ for authentication (if you need).
or docker and https://hub.docker.com/r/rocker/shiny/