I think rather than having Marketing Cloud push you the information, you are going to need to have your application pull the information. I do not know of any way to set up a trigger like this in Marketing Cloud -- I do not believe it possible, but someone else may know differently.
If you do not receive any suggestions on push this information to your application, the steps to pull the information would be as follows:
Create an Installed PackageCreate an Installed Package in the org(s) giving yourself the necessary permissions to retrieve the needed data -- potentially read on Data Extension and Emails. If you are working in an org that has a parent and child business units, you can create this installed package in the parent for use in all. If not, you'll need installed packages in each of the Orgs. This Installed Package will give you the necessary permission to retrieve Auth tokens that will be needed in step 2 for making Api calls.
Schedule some mechanism in your application to periodically make the necessary Api calls into Marketing Cloud in one of two ways to retrieve your desired data:
a) Enable Send LoggingSend Logging on your Org(s) (this must be done through a support request). Create a SendLog DataExtension in the Org(s). Make SOAP calls into Marketing Cloud to retrieve recordsDataExtensionObjects (essentially rows) from this DataExtension which are populated each time an email is sent.
b) Use Marketing Cloud's Soap Api to retrieve SentEventsSentEvents (along with ClickEventsClickEvents, OpenEventsOpenEvents, etc. if you so desire) which are created each time an email is sent.
Unfortunately, I do not believe the information you desire is accessible via Marketing Cloud REST Api which means it will need to be retrieved using the SOAP Web Service.
Marketing Cloud does also have a reporting feature that allows for the pushing of reports to an FTP server. I am not at all familiar with this feature -- I do not know how it is set up, triggered (or whether automatic or schedule triggering is even possible), nor the format of this data in the files uploaded to the configured FTP. This may be something to look into as an alternative.