Here in the UK, there are commercial products combining mobile data and GPS that will automatically generate an alert if the device is taken out of it's designated home area, and can be tracked to locate the device, and can be used by the wearer to call for help. They aren't cheap, but they aren't outrageously expensive either compared to the importance of looking after a vulnerable person. (Essentially, it's a dumbed-down smartphone and a bit of custom software.)
If you were determined to make your own equivalent, a smartphone is what I'd use to do it. I don't think it would be easy to justify the make versus buy decision, though.