For anyone else, the steps to get this working are:
Download and install PlatformIO (which is an extension for VS Code): https://docs.platformio.org/en/latest/ide/vscode.html#installation
Rearrange your folder structure to be compatible w/ what PlatformIO expects (the readme goes into more detail). https://www.dropbox.com/s/0twl3np2yw8q0iv/Screenshot 2018-10-11 17.31.50.png?dl=0
You may also need to do a bit more work to construct a PlatformIO project file (my project already came with that, so I didn't need to do this step).