Set shiny options as required for an Azure Webapp:
options(shiny.port = as.integer(Sys.getenv('PORT')).
Your custom container is expected to listen on PORT, an environment variable set by Azure.
If your image suggests EXPOSEd ports, that may be respected by Azure (undocumented behavior).
options(shiny.host = "0.0.0.0") to make your shiny application accessable to the Azure Webapp hosting environment.
shiny_opts_az()