Rest API to run User "App" and get the data

I have this App that someone in my organization created, for example:

Is it possible I can query this via rest/api/[1,2,3] to run that servlet?
There is a “Export” function to allow exporting to xls or csv which is would help, if I can’t get the data that drives the UI (format looks different).

Any help would be greatly appreciated.

Thank you!

Possible in Data Center/Server, impossible in Cloud.

You will have to expose it on your own server and handle the authentication yourself.