How can I update a third party database outside SAP after completing a Transaction?

As you need to make some changes to happen when an activity gets completed in SAP, I would suggest you to go for services. Services are the best way to communicate when you are dealing with two discrete systems.

Firstly you need to create a program on SAP side which can do the following task             

• Fetch the file from a specified directory
• Converts the file into a formatted data or send the data to a server

At the webserver end, have a listener which receives POST requests (send by SAP). You need to have proper authentication set up for this as well. Then process the file, establish a connection to the required database and perform an insert at the specified table.