Where would I find out how to programmatically pre-populate a portion of a fillable PDF?
You can download and read the documentation of the Acrobat SDK.
What I should have mentioned was, is there a .NET library I can use for this. I spent some time looking at the documentation for the Acrobat SDK but didn't see anything related to use .NET to drive the process.
Yes, you can use the "Interapplication Communication" functionality of the SDK from .NET on the desktop.