I'd like to develop an InDesign template, then try to automate it via scripting and feeding it data via a CSV or an XML file. Currently, it seems that this solution will work if I have the desktop version of InDesign.
Is it possible for a web application to take the template, populate the template with data from a database, then provide either a PDF or InDesign file to be available for download? Is there some sort of server software I'd need to have to make this happen? Or perhaps a web service I could use?
Adobe does not allow to use InDesign Desktop this way, for this you need InDesign Server.
You can create "InDesign" files without actually having InDesign, in the form of IDML. However, to "view" it or "convert to PDF" the end-user still needs InDesign. Without having InDesign, you can create IDML but you would not know how much text is going to fit in each frame and on each page. IDML is an abstract data format, and still relies on ID itself to actually 'render' the text.
Why are you considering InDesign at all? There are free PDF-backends that can translate your CSV into a PDF, such as PdfBox, and may be more suitable to your task.