To my knowledge, RStudio Connect does offer scheduling only on time & date and does not offer scheduling on a hook or alike. However, I may not know the inside out of the project.
If I am right, that would be a great feature request. 
Here is a quick thought on this. One way would be to deploy a new content to start the running of the app like a schechuling.
There is several ways to deploy to rstudio connect, including a programmatic way and a Git-based way. That means at the end of your program to check if data refresh is complete, you may be able to start a deployment programmatically by sending the source of your app to connect for a new deployment. This programmatic deployment could be done by your other program or using a CI/CD system if you have one.
Otherwise, you may tag your git repo for a new version and that would schedule a new deployment if you configure connect that way for git deployment.
Hope it helps. And let's see if someone have a better solution ! 