I have a bit of challenge. Knowing only the basic URL for the sharepoint installation, can I get a list of the site collections that have been created using only the basic web s
Unfortunately, no. That functionality isn't available from the out-of-the-box web services. The only option that might work is using the Search Web Service (I'm not familiar enough with it to know).
This walkthrough describes how to add your own custom web services to the product. I strongly recommend this approach as it's very likely there will be other missing functionality you will need to add - if not now, in the near future.