@Joris Meys 's answer covered the topic of splitting shiny code into files. Though one part of the question is to use the run app button, which may not be available even if the organization make a valid shiny app.
If you search this question you can find this issue, then following to the commit made in that issue you can find what's the rule to have a run app button, and this function about isShinyAppDir, this function of shinyfiletype:
Basically any folder have ui.R, server.R, app.R, global.R, www folder will be considered as shiny folder(the detailed conditions are more complex, see source code), then the above 4 files will have run app button.
One thing I noticed is that usually you can keep the app running, make some changes then reload app to see the changes, but if you sourced other file, reload app button will not reload changes in that sourced file.
