I think this depends on how your Shiny app works. Even with a connection to a database, an app might load all data on deployment. In this set-up, that database query stage is only handled once, and updating or adding data requires redeploying. Based on your description of your problem, that sounds like what's happening here.
One option: If this is an interactive document, you can use RStudio Connect to schedule a regular re-running, updating your data in the process and saving older versions of the report (scheduling R Markdown documents in Connect).
Another option is to update how your shiny-app interacts with data. Shiny applications can be written to dynamically pull data from a SQL database or other API. Intro doc on this, "Shiny - Database basics - dplyr and DBI".
The shiny packpage pool might help