There is really no difference between dealing with another website and dealing with a desktop application when it comes to Web APIs.
Basically, what you'll want to do is design something where they are able to make calls to a specific page and send specific information. Then you take the information, process it and do whatever, and then send some sort of response.
You'll want to figure out what functions you'll need to accompany so they can send and receive the appropriate information (getLogActivity, saveLogActivity, etc.).
If you haven't yet, you may want to grab a book or two on RestFUL Web Services. This book from O'Reilly is a bit dry (as are all of their books), but pretty useful on the topic (they also discuss other types of Web Services, like SOAP). RESTful Web Services - O'Reilly Media
Hope that helps.