Is it possible to imbue (via C# code) location self-awareness into a small piece of "Gadgeteer" (http://gadgeteer.codeplex.com/) hardware?
IOW, I'm wondering if it's possible to write C# code to download into a widget (embedded device) that will be able to use the Geolocation class (or so) to know its own coordinates and then send a wireless signal (using sockets, for example) to a Windows 8 app running on a tablet.
Yes. You can use the GPS module to glean location (at least outside) and the Wi-Fi module (or any of the connectivity modules) to facilitate a connection to a server and post the result. The Win8 tile app could then read the Gadgeteer module's location from the same web server.